Gambling Math

Gambling math is an interesting beast, to say the least. For most recreational gamblers, they refuse to even touch it and avoid it like the plague. Why?

For most people, they either have a deep-seeded hatred for math, or they just haven’t had it explained to them properly. While we fully understand this, it can create some issues with fully understanding what you’re doing and what you’re betting on.

Why You Should Learn the Math

The big question we get asked a lot is whether or not you can survive in the gambling world without knowing any of the math behind what you are doing. The answer is as it is with most things in gaming, yes and no. Can you still make bets and win without knowing any of the math behind your bets? Of course, you can. Does it benefit you to learn the math enough to warrant learning it? The answer here is also yes.

If you’re a table games player, learning the math behind gambling can help you to make the smartest bets possible and give yourself the best odds to win. If you’re a professional sports bettor or an amateur aspiring to be one, you MUST learn this math. It is the key to knowing WHAT you’re betting and whether or not you are making a profitable bet.

The answer to the big question is that if you’re a recreational gambler playing table games you can get away with not learning this, but it will greatly benefit you to do so. If you are a professional sports bettor or an aspiring one, this math is a must learn if you have any hopes of having a sustainable career in sports betting.

Calculating Payouts Based on Odds

The most essential thing that you’re going to need to know about gambling math is how to calculate the amount you are supposed to be paid out. This is the first key to knowing whether or not you are making a good bet and whether or not the risk is worth the potential reward. It’s also important to make sure that you get paid correctly.

While most online sites will do this for you automatically, you’ll need to know this if you bet in a brick and mortar location, bet with friends, or want to make sure that the online casino or sportsbook didn’t make a mistake with your bet. It’s also going to be extremely important to calculate into your research to help you make smart bets regardless of where you are betting.

The bottom line is that this is a must learn. We will do our best to make the process as painless as possible.

American Odds

These are sometimes referred to as moneyline odds and are the odds you’re going to see most commonly in the United States. While most casinos online will allow you to convert the odds to whichever you like, this is usually the starting point especially for casinos servicing the United States.

American Odds will look something like this:

The plus or minus sign indicates whether or not the team/player is a favorite or underdog and the number indicates by how much. A plus sign indicates a team that is an underdog and a minus sign indicates a team that is a favorite. The larger the absolute value of the number, the larger the favorite or underdog.

Calculating your payouts will be slightly different for underdogs than favorites. We’re, of course, going to walk you through both.

For underdogs, the number shown will be the amount you would get paid for betting $100. So, if the team is +300, you would profit $300 for every $100 you bet. This means that if you divide the odds by 100, you will get the amount of money you will earn for each dollar wagered. So, +300, you will get paid $3 for every $1 you wager.

Here is the simple formula for calculating the payout. The above paragraph was just so you understand where this formula is coming from.

For example, let’s say you want to bet $50 on a team to win that is +250.

• Profit = $50 * (250/100)
• Profit = $50 * (2.5)
• Profit = $125

As a second example, just to make sure we’re clear, let’s say you want to bet $300 on a team to win that is +400.

• Profit = $300 * (400/100)
• Profit = $300 * (4)
• Profit = $1200

For favorites, the formula is going to be a little different when to calculating your payout.

For example, let’s say you want to bet $80 on a team to win that is -180.

• Profit = $80 / (180/100)
• Profit = $80 / (1.8)
• Profit = $44.44

There are a few things that need to be pointed out here to make sure you have a complete understanding. First, you always use the absolute value of the odds in the calculations. This means that you basically ignore the minus sign and treat it as a positive number. Notice how in the above example we didn’t use negative 180, but we just used 180. This is always the case.

Secondly, this is the calculation of profit, not of the total payout. When the casino or sportsbook goes to pay you out, they are going to give you the potential profit plus your initial wager. So in the last example above where we are betting $80 at -180, our potential profit is $44.44, but that is not the amount the casino or sportsbook would pay us out on the bet. If we gave them $80 and they only gave us $44.44 back for a winning bet, we would be losing money.

They give you back the $44.44 in profit plus the initial $80 from your wager. So the total payout is going to be $124.44. Just make sure that when you’re looking at betting that you don’t get confused and think you are profiting $124.44.

Decimal Odds

Decimal odds are what you’re most likely going to find in Europe and to be honest, they’re the easiest to work with in regards to calculating potential payouts. As more and more people realize how easy they are to work with, they are becoming more and more standardized. One of the biggest perks of using decimal odds is that you don’t have to use different calculations for favorites and underdogs. Both are figured exactly the same with a much simpler formula.

It’s that simple. You just have to take the odds given and multiply that by how much you are betting. For example, let’s say you are betting $150 on a team to win that is 2.4.

• Payout = Stake * Odds
• Payout = $150 * 2.4
• Payout = $360

Now, if you’re paying attention, you see that we are calculating the payout and NOT the profit here as we did in the previous example. This number is the total of your profit as well as the original wager that is going to be returned. To figure out potential profit, you have to make one small adjustment. There are two ways of doing this.

The first way is altering the formula as such:

So in our above example, here is what our potential profit would look like.

• Profit = $150 * (2.4 – 1)
• Profit = $150 * (1.4)
• Profit = $210

The other way you can do it is to calculate using our total payout formula and then subtract out your initial wager.

• Profit = ($150 * 2.4) – $150
• Profit = ($360) – $150
• Profit = $210

Fractional Odds

Fractional odds are most popular in the United Kingdom and can be a little more confusing to work with if you aren’t a big fan of math. That being said, they still follow a simple formula that will help you to calculate your payouts and profits fairly easily.

Effectively the formula is the exact same as it is for decimal odds except instead of the odds being a decimal, they are a fraction, and it gives you your profit instead of total payout. Let’s look at the exact same example as above (we’ll explain why in a second).

You are betting $150 at odds of 7/5.

• Profit = $150 *(7/5)
• Profit = $150 * (1.4)
• Profit = $210

If you want this to be your total payout, just add back in your initial wager. Now, for those of you that don’t remember much about fractions, here’s a quick refresher. To “solve” the fraction and convert it to a more user-friendly number, you take the top number (the numerator) and divide it by the bottom number (the denominator). So in the above example, we took 7 and divided it by 5 to get 1.4.

If you recall our formula for decimal odds profit, it was the following:

We will get into how you figure this out later, but 7/5 is the exact same odds as 2.4. Notice that if you take 7 and divide it by 5, you get 1.4. In the above formula for decimal odds, 2.4 – 1 also equals 1.4. We used the same example so that we could point out to you that the profit formulas are relatively the same idea.

A Convenient Chart

Here is a chart of the profits and payouts you would receive for a $10 wager.

Converting Odds to a Different Format

As we’ve already covered, odds on bets are offered in three different formats – American, Decimal, and Fractional. We’ve covered how you calculate your potential profits and payouts with these, but now we’d like to talk about how you convert one format to the other two. The reason this is important is that it makes sure you aren’t comparing apples to oranges with two different bets or line shopping.

For example, which is a better bet?

• Chicago Bears to Win -150
• Chicago Bears to Win 1.59

If you’re a math wizard and already know how to do these calculations, then you’re going to be able to figure this out. For the rest of us peons, it’s going to be real tough to look at these two different bets and decide which is best. For those taking a guess, the first bet is the better of the two bets.

The Easy Way

This section would not be complete if we didn’t put up the easy solutions to converting odds to a different format. There are two resources that are going to help immensely in the conversions if you don’t want to do them by hand. The first is that a lot of good betting sites will allow you to convert the odds with a click of a button to whichever format you prefer. Some sites will require you to do this manually for every bet while the better sites will allow you to change all of the odds into the same format with one click of the mouse.

The second way to calculate these easily is using an odds converter. This is a tool that will convert your odds into all the different formats for quick reference. We have one built that you can try out and use for free. Ours does go a step further as well and give you implied probabilities which we will be going into later in this guide.

American Odds to Decimal Odds

Converting American odds to Decimal odds can be done with the following formulas. Depending on whether or not the odds are for favorites or underdogs, determines which formula you will use. The two formulas are as follows:

• If Odds > 0 (Underdogs), then (Odds + 100) / 100
• If Odds

*Use the absolute value of the odds. This means to ignore the negative sign and treat the odds as a positive number.

Let’s convert +150 to decimal odds.

Since +150 > 0 and the team is an underdog, we use the following formula:

• Decimal Odds = (Odds + 100) / 100
• Decimal Odds = (150 + 100) / 100
• Decimal Odds = (250) / 100
• Decimal Odds = 2.5

Let’s convert -220 to decimal odds.

Since -220

• Decimal Odds = (Odds + 100) / Odds
• Decimal Odds = (220 + 100) / 220
• Decimal Odds = (320) / 220
• Decimal Odds = 1.4545

American Odds to Fractional Odds

Converting American odds to Fractional odds can be done with the following formulas. Depending on whether or not the odds are for favorites or underdogs, determines which formula you will use. The two formulas are as follows:

• If Odds > 0 (Underdogs), Odds/100
• If Odds

Let’s convert +150 to fractional odds.

Since +150 > 0 and the team is an underdog, we use the following formula:

• Fractional Odds = Odds/100
• Fractional Odds = 150/100
• Fractional Odds = 3/2

Let’s convert -220 to fractional odds.

Since -220

• Fractional Odds = 100/Odds
• Fractional Odds = 100/220
• Fractional Odds = 5/11

Technically, 150/100 and 100/220 are fractional odds that are correct, but are just not simplified. Simplification of a fraction is the process of getting it into its smallest and easiest to work with format. 150/100 is the exact same number as 3/2. If you don’t believe us, divide them out into decimals (150 divided by 100 and 3 divided by 2), and you will get the exact same number.

To simplify a fraction, you divide the top and the bottom by numbers that evenly go into both. For example, let’s simplify 150/100.

Both numbers are divisible by 10 so we will divide both by 10.

We continue doing this until there are no more numbers that will go evenly into both the numerator and the denominator. 15 and 10 are both divisible by 5, so we divide both by 5.

There are no numbers that evenly go into 3 and 2, so this fraction is considered simplified.

Decimal Odds to American Odds

As usual, the American/moneyline style odds are a bit more confusing to work with. Though a little more confusing, they’re still fairly easy to work with and convert to. Depending on whether or not the odds are for favorites or underdogs, determines which formula you will use. With decimal odds, favorites are less than 2.0 and underdogs are greater than 2.0. The two formulas are as follows:

• If Odds > 2 (Underdogs), (Odds – 1) x 100
• If Odds

Let’s convert 2.5 to American odds.

Since 2.5 > 2 and the team is an underdog, we use the following formula:

• American Odds = (Odds – 1) x 100
• American Odds = (2.5-1) x 100
• American Odds = (1.5) x 100
• American Odds = 150

A plus sign is added to the front for formatting.

American Odds = +150

Let’s convert 1.4545 to American odds.

Since 1.4545

• American Odds = 100 / (Odds -1)
• American Odds = 100 / (1.4545 – 1)
• American Odds = 100 / (.4545)
• American Odds = 220.02

A minus sign is added to the front for formatting.

American Odds = -220.02

Decimal Odds to Fractional Odds

Converting decimal odds to fractional odds can be done with the following formula. It does not matter whether or not the team is a favorite or not when working with this conversion.

Let’s convert 2.5 to fractional odds.

Since it does not matter whether the team is a favorite or not, we use the following formula:

• Fractional Odds = (Decimal Odds -1) / 1
• Fractional Odds = (2.5 – 1) / 1
• Fractional Odds =1.5/1

We do not want to have any decimal points in our fraction, so we’re basically going to do the opposite of simplifying as we did in the earlier example. Instead of dividing the top and bottom number by the same number, we’re going to multiply it by the same number. The easiest way to do this is to figure out how many decimal places you need to get moved. In the above example, we need to move the decimal point one space to get a whole number on the top.

• If we have to move the decimal by one space, multiply by 10.
• If we have to move the decimal by two spaces, multiply by 100.
• If we have to move the decimal by three spaces, multiply by 1000.

This continues on and on how you would assume. So in our above example, we multiply the top and the bottom by 10.

We then simplify this fraction as we did earlier by dividing the top and the bottom both by 5 which gets us 3/2.

Let’s convert 1.45 to fractional odds.

Since it does not matter whether the team is a favorite or not, we use the following formula:

• Fractional Odds = (Decimal Odds -1) / 1
• Fractional Odds = (1.45 – 1) / 1
• Fractional Odds = (.45) / 1

Again, we need to move the decimal place to get whole numbers. Since we need to move it four spots (to make .45 into 45), we multiply both the top and the bottom by 100.

Again, we need to simplify this fraction because it is still divisible by the same number. We see that 5 goes into both numbers, so we divide the top and the bottom by 5. This gets us 9/20.

Just as a side note, the actual decimal odds that correspond to the examples we’ve been using are 1.4545454545 repeating forever, so the actual fractional breakdown is 5/11. We just simplified here, and for all intents and purposes, that works just fine for us. We just wanted to point that out in case you were calculating a different way and got a slightly different number.

Fractional Odds to American Odds

When converting fractional odds to American odds, it again matters whether the bet is for a favorite or for an underdog. Determining which is which is done by looking at which number is greater in the fraction. If the top number (the numerator) is greater, then the team is an underdog. If the bottom number (the denominator) is greater, then the team is a favorite. The two formulas are as follows:

• If numerator > denominator (the team is an underdog), Odds x 100
• If denominator

Let’s convert 3/2 to American odds.

Since 3>2 and the team is an underdog, we will use the following formula:

• American Odds = Odds x 100
• American Odds = (3/2) x 100
• American Odds = (1.5) x 100
• American Odds = 150

A plus sign is added to the front for formatting.

American Odds = +150

Let’s convert 5/11 to American odds.

Since 5

• American Odds = 100/Odds
• American Odds = 100 / (5/11)
• American Odds = 100 / .4545)
• American Odds = 220.02

A minus sign is added to the front for formatting.

American Odds = -220.02

Fractional Odds to Decimal Odds

Converting fractional odds to decimal odds can be done with the following formula. It does not matter whether or not the team is a favorite or not when working with this conversion.

Decimal Odds = (Fractional Odds) + 1

Let’s convert 3/2 to decimal odds.

Since it does not matter whether the team or bet is a favorite or not, we use the following formula:

• Decimal Odds = (Fractional Odds) + 1
• Decimal Odds = (3/2) + 1
• Decimal Odds = (1.5) + 1
• Decimal Odds = 2.5

Let’s convert 5/11 to decimal odds.

Since it does not matter whether the team or bet is a favorite or not, we use the following formula:

• Decimal Odds = (Fractional Odds) + 1
• Decimal Odds = (5/11) + 1
• Decimal Odds = (.4545) + 1
• Decimal Odds = 1.4545

A Convenient Chart

If you’re practicing your conversions, this chart should help you have plenty of practice. If you don’t want to do the math and just want to see the conversions, this chart will also be of big help to you. Keep in mind that a few of the numbers are rounded off so your calculations will be extremely close, but may be a tiny bit different. This really should make no difference, though, in the grand scheme.

Probability vs Odds

Something we want to breakdown for you is the difference between probabilities and odds. Probability, by definition, is the likelihood or chance that something will happen. Odds, by definition, are the ratio of a player’s chances of losing to his or her chances of winning. While these might sound like the same thing, they are in fact different, yet they represent the same thing.

Probability

Probabilities are represented as a fraction, percentage or proportion between 0 and 1. For example, let’s say there are 10 tickets in a drawing and you have 1 of the tickets. Your probability of winning can be presented as 1 in 10, 10%, or 0.1.

Odds

Odds are represented as a ratio. Continuing with the above example, your odds would be represented as 9 to 1. You have 9 opportunities to lose to your 1 opportunity to win. Converting odds to probability is easy and uses the following formula:

So, our numerator would be 1, and the denominator would be 9+1 or 10. Our probability is 1/10 or 10%.

The Use of Odds in Gambling

In gambling conversations and situations, the term “odds” is not really used to mean the actual chance of winning. It is typically used as a subjective estimate of the odds rather than a pure mathematical computation of the odds. What does this mean?

In reality, though, the dog is actually less than 20% likely to win the race. This is because when the sportsbook is talking about the odds, they are talking about what they are going to pay you on a winning bet. This is different than the true odds which would be how likely the dog is to win the race. What they’re actually saying here is that they are going to pay you $4 for every $1 you bet.

“Good Odds”

When a sports bettor says that the odds presented on a bet are good odds, they are referring to the relationship between the sportsbooks odds and the true odds. We can’t ever know the true odds of the dog winning the race, but most likely it’s going to be less than the payouts offered. The posted odds are usually going to overestimate the dog’s chances of winning. This makes sure that the bettor is underpaid on their wager, giving the sportsbook the opportunity to make a small profit.

Here’s a better example where we actually know the true odds of something. Let’s say you are playing roulette and want to bet $10 on black. The house tells you that you are going to be paid odds of 1 to 1 on your black bet. If we convert this to probability, our numerator (winning chances) is 1, and our denominator (total possibilities) is 2. Our probability is 1/2 or 50% on these odds.

If this were a fair bet for us, our true odds of winning the bet would be the same as the posted odds. A roulette wheel has 38 slots on it, and 18 of them are black. We remember our odds are a ratio of opportunities to lose to opportunities to win. So the true odds of rolling black are 20 to 18. Converting this to probability, we see the true probability of rolling black is 18/(20+18), which is 18/38 or 47.37% to win.

As you can see we are getting paid out as if the spin is 50% likely to happen, but it is in fact only 47.37% likely to happen. The posted odds are overestimating your chances of winning to ensure that you are underpaid for your win. Why? This is how the casinos make their money. This is the house advantage.

Combinations

Taking ourselves back to math class again, we need to talk about combinations and computing the probabilities of individual events happening in combination. Let’s look at an example to illustrate this.

Let’s say you are playing a slot machine with three reels and 4 different symbols on each reel. Let’s also say that each reel is identical and one of the four symbols on each reel is a gold star. If you get all 3 gold stars to line up on a spin, you win the jackpot. What is your probability of winning the jackpot?

Well, first let’s look at the probability of getting a gold star on each reel since they all spin independently. The probability of getting the gold star on reel one is 1 in 4 or 25%. The probability of getting the gold star on reel two is also 25%, and the same is true for reel three. A lot of people would then assume that their probability of winning the jackpot is 25%.

The problem is that each reel is independent, so you need the 25% occurrences to all happen at the same time. The probability of events occurring in combination is always going to be less than the probability of each of the events occurring independently. The way to calculate the probability of the occurrences in combination is to multiply the probabilities together.

So to find this, we multiply the following:

To convert this to a percentage, we solve the fraction by dividing 1 by 64 and then multiplying by 100. This gives us a probability of winning the jackpot of 2.77%.

Combinations Lead to Overestimation of Winnings

Let’s take our slot example a step further to show you why people have a tendency to think they have a better shot of winning a jackpot than they actually do.

Let’s say now that our three reel machine has 25 symbols on each reel and 1 of those 25 on each reel is the gold star. If you get 3 gold stars in a row at the same time, you win the jackpot.

The odds that you will see at least 1 gold star on a spin are calculated by taking the probability of seeing it on one reel and multiplying it by the number of reels which is 3.

You have a 12% chance to see a gold star on every single spin.

The odds of getting two gold stars on the same spin is calculated by multiplying 1/25 * 1/25 which gives us a 0.16% chance of spinning two gold stars. This means that you will get two gold stars once every 625 spins of the wheel. That’s fairly rare, but definitely, something that you’re going to see. Because of this, people are likely to think that they have a better shot of winning the jackpot than they actually do. They assume that when they get two, they are really close to winning. But, let’s calculate the true odds of winning the jackpot to see how close they really are.

The odds of getting three gold stars is calculated by multiplying 1/25 * 1/25 * 1/25 which gives us a 0.0064% chance of hitting the jackpot. This means you will hit the jackpot 1 every 15,625 spins. As you can see, two gold stars are not very close to three, but in your mind, you may think that it is.

Implied Probability / Finding Value

Implied probability is a term you will hear a lot in the gambling world. It basically means exactly what we have been talking about in terms of probability. It is the percentage chance outcome of an event as represented by the payout odds. Comparing this to the true odds of an event will let you know if you have found a good value bet or not.

How do you know the true odds of something happening? Well, you have to figure those out yourself. Typically, sports bettors will have mathematical formulas that they come up with to determine how likely they think someone is to win a game or for some event to happen. It is up to the bettor to develop this formula and figure out which statistics are the most important.

Once you calculate what you think the true odds are of something happening, you compare it to the implied probability of the bet being offered by the sportsbook. If the implied probability is less than what you have figured for the true odds, the bet has value. Let’s look at an oversimplified example.

Let’s say we are going to be on a dog race that has only four dogs in it. These are the odds presented by the sportsbook:

• Lady In Red +600
• Sherman’s Tanks +300
• Dinglehopper +300
• Ace Hole +200

We come up with a formula to calculate how likely we think a dog is to win the race. Our formula assigns points for a few different things, and then we convert the points into what we think the true odds are of that dog winning the race.

Remember, this is not a real formula and just an example. Let’s say that we think the dog’s top speed recorded is 50% important, how many wins they have is 25% important and how many races they’ve run in the last month is 25% important. For races run, less is better.

Here are the profiles we have on the dogs:

Lady In Red

• Top Speed: 38 mph
• Wins: 3
• Races Run Recently: 9

Sherman’s Tanks

• Top Speed: 40 mph
• Wins: 10
• Races Run Recently: 11

Dinglehopper

• Top Speed: 40.2 mph
• Wins: 9
• Races Run Recently: 8

Ace Hole

• Top Speed: 42 mph
• Wins: 11
• Races Run Recently: 15

Our formula for calculating what we think the true odds are is as follows:

Dogs are given rankings with 4 points to the top dog in the category, 3 to second, 2 to third, and 1 to last.

• Lady In Red = (.5*1) + (.25 * 1) + (.25 *3) = 1.5
• Sherman’s Tanks = (.5*2) + (.25 *3) + (.25 *2) = 2.25
• Dinglehopper = (.5*3) + (.25 *2) + (.25 *4) = 3
• Ace Hole = (.5*4) + (.25 *4) + (.25 *1) = 3.25

Now that we have our points system calculations finished we need to turn these into probability percentages to see what we have calculated is likely to happen in the race. To do this, we will calculate the percentage of points each dog has of the total points we awarded.

Now we calculate what percentage of the total each dog’s score is:

• Lady In Red = 1.5/10 = 15%
• Sherman’s Tanks = 2.25/10 = 22.5%
• Dinglehopper = 3/10 = 30%
• Ace Hole = 3.25/10 = 32.5%

These percentages represent our feelings on what is going to happen in the race. They are our true odds/probability predictions on the race. We must now calculate the implied probabilities of the payout odds posted and see if we can find some value to bet.

• Lady In Red +600
• Sherman’s Tanks +300
• Dinglehopper +300
• Ace Hole +200

To calculate the implied probabilities, we use the following formula:

So first, we must convert all of our American odds into decimal odds. We covered this earlier in this article, so please scroll up if you need a refresher. We will go ahead and just convert them here for you.

• Lady In Red +600 14.29%
• Sherman’s Tanks +300 25.00%
• Dinglehopper +300 25.00%
• Ace Hole +200 33.33%

One thing you may notice is that the totals here do not add up to 100% as they did with our true odds/probability calculations. This is because the sportsbook is taking the difference as their margins.

Let’s compare our implied probabilities to our true probability predictions. Remember, if we find a bet that has an implied probability lower than the true odds we predicted, then we would say the bet has value. In simpler terms, the sportsbook is going to pay you more money for a dog that has a less likely chance to win. So if the sportsbook is paying you as if the dog only has a 20% chance to win, but you think it has a 30% chance, you are getting paid much more on that win than you “should be.” Technically, you should be getting paid 30% chance rates for the dog that you think has a 30% chance rate.

A dog with a 30% implied probability will be paid out by the sportsbook at American odds of +233.33. A dog with a 20% implied probability will be paid out by the sportsbook at American odds of + 400. This means that if you bet $100, you would be getting paid $233.33 in profit on the first dog and $400 in profit on the second dog.

The second dog is getting paid out as if it has a 20% chance of winning or that it will win 2 out of 10 races. If you think the dog will win 3 out of 10 races (or has a 30% chance of winning), you’re going to see a profit on this bet if all of that is true. Let’s look at what happens if we bet $100 on 10 different races. We’re going to assume for argument’s sake that the dogs are going to hit their exact true odds and the house is taking no money.

If you’re getting paid out at 30% (+233.33) rate for a dog that you think will win 30% of the races:

Race	Wager	Result	Profit/Loss
1	$100	Win	$233.33
2	$100	Win	$233.33
3	$100	Win	$233.33
4	$100	Lose	($100)
5	$100	Lose	($100)
6	$100	Lose	($100)
7	$100	Lose	($100)
8	$100	Lose	($100)
9	$100	Lose	($100)
10	$100	Lose	($100)

Total Profit = $0

Because the implied probabilities and the true odds/probability (what you think is going to happen) are the same, the bet is even, and you are expected not to win or lose any money.

But, as we’re trying to point out if you see a situation where you think a dog is 30% to win and is being paid out with an implied probability of less than that (20% in our example), you have the chance to make some profit. Here is that that would look like over the 10 races.

If you’re getting paid out at a 20% (+400) rate for a dog that you think will win 30% of the races:

Race	Wager	Result	Profit/Loss
1	$100	Win	$400
2	$100	Win	$400
3	$100	Win	$400
4	$100	Lose	($100)
5	$100	Lose	($100)
6	$100	Lose	($100)
7	$100	Lose	($100)
8	$100	Lose	($100)
9	$100	Lose	($100)
10	$100	Lose	($100)

Total Profit = $500

As you can see now by example, finding a bet where the implied probability is lower than the true probability that you predict could result in a value opportunity for you to make some money.

Let’s get back to our current example and compare the implied probabilities with what we calculated

Dog Name	Posted Odds	Implied Probability	Our Probability Predictions
Lady In Red	600	14.29%	15%
Sherman’s Tanks	300	25.00%	22.50%
Dinglehopper	300	25.00%	30%
Ace Hole	200	33.33%	32.50%

As you can see in the above example, Lady In Red and Dinglehopper have implied probabilities below our predicted probabilities. Technically, if you are correct in your predictions, you will see value from betting these dogs. Does that mean you are guaranteed to win money on this race? No, it does not. But if you constantly make bets where you have value, you will make money over the long run.

Technically by your calculations, your dogs are only going to win the race 45% of the time, and you will lose 55% of the time. But, when you win, you will make enough to cover the losses and turn a profit. Again, this is assuming that your calculations are correct and that you have enough opportunities to make these bets to see the value come through. If this is the only race you are ever allowed to bet on, you might lose and never get a chance to realize your value.

Thankfully, if your formula is correct and picks winners, you’re going to be able to use it on any and all dog races and clean up. What’s the secret to developing a good formula? That’s up to you to figure out. Start by determining the statistics that you have access to that you think are important and begin assigning weights to them. You can follow the above process for any sport, and professional bettors do just that. The only differences would be what sport they are doing it for and the complexity of their formula. Just make sure you apply weight to each category and then plug in your numbers and convert to percentages.

A Convenient Chart

Here is the chart from earlier with the different odds converted. This time we have included the associated implied probabilities as well.

Gambling Fallacies

A discussion about gambling math is never complete without discussing the fallacies that develop in people’s minds about gambling. Many people think there are systems to beat the casino and things of that nature that can help give them a leg up. The reality is that math is supreme and cannot be circumvented. With some forms of betting (specifically sports betting), you are dealing with people’s predictions of the math and a few other factors that make being profitable long term possible.

Casino games with a house edge, though, the best you can do is make decisions based on the math to increase your chances of winning and lower that house edge. Let’s take a look at some of the more popular misconceptions that arise in the gambling world by uninformed individuals.

Everything Will Even Out

This is the basis of the gambler’s fallacy which is the belief that things are going to even out. For example, a coin has the exact same chance of landing on heads as it does tails. There is no memory device in the coin, and each time it flips it has a 50% chance of landing on either side no matter what has happened before. Here’s a question to demonstrate this. If I flip a coin and it lands on heads six times, what are the chances that it will land on heads the next flip?

Surprisingly, a lot of people think it is less likely to land on heads which is simply not true. As this is easily the most popular and prevalent fallacy, here is a complete write up on the topic and one we highly recommend checking out.

Randomness Contains No Patterns

Random events are exactly what they say they are – completely random with each event being completely independent of the others. That being said, there are times that patterns will arise in random events. The important takeaway, though, is that these patterns have no extra likelihood to continue.

For example, if you are playing roulette, each spin is completely random. If the wheel comes out with three reds, then three blacks, and then three reds again, people are likely to think that three blacks are more likely to come up next due to the apparent pattern. While this is definitely a pattern thanks to fundamental uncertainty, it has no bearing on the next spin of the wheel.

The additional problem that arises out of this is that people will start to believe that something is in fact, not random due to the presence of patterns. This will cause people to start believing that things are rigged or not random just because they found a pattern. Again, this is simply not the case.

A Commonly Occurring Event Equals a Bias

Let’s say you are playing roulette and the number 11 keeps coming up over and over again. A lot of people are going to start to believe that the wheel has a bias towards the number 11. 99.99% of the time this is not going to be the case. Our brains are just tricking us to ignore the probabilities and the fact that things like this do happen.

Self-fulfilling prophecy can create additional issues here. Once you start looking for the number 11, you’re going to start noticing when it comes up more and more, and it’s going to feel like it’s coming up more likely than it actually is. Even if it is coming up more often, it does not mean that the wheel has a bias towards that number. This is just how statistics and randomness work sometimes.

While we were hesitant to say it because we don’t want to feed the fallacy, we did mention that this was the case 99.99% of the time. The reason we said this is that you do need to be careful if you are ever gambling or doing something outside of a casino or regulated environment. For example, if you were betting on coin flips in the casino you would have nothing to worry about because you would know that the coin was fair and regulated by the gaming commission and third party auditors.

If you were betting with someone on the street, though, you have no way of knowing if the coin is biased (or in other terms rigged).

Players Can Gain an Edge Over Randomness

What is interesting is that the reason people love gambling is that they feel they have some control over the outcome of the games. For example, have you ever watched a rerun of a live sporting event that you knew had already ended? Does it just not feel the same even if you don’t know the outcome? This is because you subconsciously feel like your cheering and rooting in some way, shape, or form affects the outcome of the game.

We all know logically (hopefully) that it does not, but we still get the feeling that it does. This is the same reason that people like gambling. If it was strictly about the money and you felt you had no control over what was going on, you could just give your money to someone else and let them go bet for you. The thing is, though, you don’t want to do that. That’s partially because gambling is fun but also because if we’re honest with ourselves, we do feel like we have some effect on what happens.

This is why we say we are lucky or unlucky. We think that our personal luck factor is affecting the outcome of the game or bets. The point here is that if you are playing a random game of chance, you have no control over the randomness at all. Taking this a step further, you also have no ability to predict the future random occurrences. They are called random for a reason.

The fun part about this one and all of these fallacies for that matter is some of the smartest minds in the world believe some of these untruths. You’d be surprised how many people believe they have some sort of control over the random events or have some sort of system for beating the casino. Remember, if they truly had a system that worked, they would quit their job and be one of the richest people in the world.

Law of Large Numbers

This is a mathematical concept that is imperative to understand in order to get a full grasp of long term profitability or long-term loss as it pertains to gambling. By definition, the law of large numbers states that as the sample size increases, the average of the actual outcomes will more closely approximate the mathematical probability.

Let’s unpack this as it’s considerably simpler than it sounds but it still very important. It’s basically saying that the more times that something is done, the closer the average will get to “what it is supposed to be.” For example, let’s look at our classic coin flip example. If we flip a coin 1 time, it can be heads or tails. If we flip it 10 times, it could bet 10 heads, 10 tails, or a mix of the both. If we flip it 100 times, it could be 100 heads, 100 tails, or a mix of both.

According to the law of large numbers, though, the more times that we flip the coin the close it’s going to get to be 50% on each side of the coin. Technically, it is not the actual number of flips that gets us closer to the probability percentage but the average number of flips. We will leave that one for another day, though. The important part here is to realize that the more times that something occurs, the closer it is going to be to the “correct probability.”

How does this affect us with gambling? Let’s say you are a poker player who plays heads-up tournaments. These are tournaments where you play against one other player, and the winner takes all the money.

You actually can’t tell at all. With such a small sample size, it’s easy for the variance to be skewing who the better player really is.

Let’s say these players play 5,000 tournaments and player A wins 3,000 of them. Now you can more confidently say that Player A is the better player. It’s entirely possible for Player B to win more in the short term but as the law of large numbers states, the larger the sample size (the more times the tournament is run), the closer to the mathematical probability we are going to get.

The point is that you don’t want to be too results oriented from a small sample size and change something that isn’t working just because you are losing. For example, if you have a formula for betting basketball and you lose your first five games, it does not mean your formula is bad at all. You’ll need a much larger sample size to figure out if your formula is working or not. This is why we recommend betting small amounts initially or applying your formula to past games to test its validity.

Win Rates

What is a surprisingly easy calculation, is one that is often ignored or incorrectly calculated by professional sports bettors and skilled game-players. This section is specifically targeted at players playing those games (sports betting, poker, skill games, etc.) that are proven sustainable for long term profit. This section really does not pertain to those playing games where the casino has an edge unless you’re just curious.

Your win rate is essentially how much money you are making. What it does (or what it should do), though, is goes a step further and calculates what you are making per hour as well as calculates for your expenses. Too often, bettors will incorrectly calculate their win rate and leave themselves in a world of hurt when they realize the hard way that they aren’t making as much money as they thought.

A lot of times, people will calculate this to determine if they can leave their “real job” and pursue betting full time. If you are planning to do this, great, but please remember one thing.

We cannot stress this enough, nor can we point out how many times we have seen people lie to themselves when calculating this.

Let’s talk about how to calculate this the correct way. First, you need to have accurate data. You need to know the following information, and it needs to be accurate.

For sports bettors, this is going to include ALL of your research time and all of the time you spend watching sports shows or seeking advice from friends and colleagues. Why is this so important? This is important because most people have a tendency to overestimate their hourly rate because they don’t include all the time they spend on something. For example, if you calculate that you are making $1000 a week playing poker and you play 40 hours a week, you may be excited to see that you are making $25 an hour ($1000/40 hours).

The problem is, though, that you forgot to calculate in the 20 hours you spent studying and the 5 hours of total travel time to and from the casino. Your actual hourly win rate is $15.38 ($1000/(40+20+5). While this may still be sufficient for you, it’s significantly lower than what you had calculated.

The key here is to get the most honest picture possible. This is not something that you have to share with other people or ever make public. It is strictly to help you make the best-informed decisions about your betting career.

To calculate your specific win rate per hour, here is the formula you should use:

Let’s look at an example to see how this might be calculated. We’ll use a professional sports bettor looking for their hourly win rate for the month. Here are the stats we have to work with.

Profit Numbers

• Total Money Won: $14,350
• Total Money Lost: $11,280

Expenses

• Research software: $20 per week
• Sports package subscription: $19 per month
• Travel to casino: $0 (Betting online)

Time Commitment

• Research: 25 hours per week
• Game watching: 12 hours per week
• Discussions with bettors: 10 hours per week

We are assuming a month is 4.5 weeks.

• Hourly Win Rate = {($14,350 – $11,280) – [($20*4.5) + $19]}/ (25 + 12 + 10)*4.5)
• Hourly Win Rate = {($3070) – [(90) + 19]}/211.5
• Hourly Win Rate = $2961/211.5
• Hourly Win Rate = $14

This sports bettor is making $14 an hour for their efforts.

Extrapolation Concerns

The biggest mistake people will make with their win rate is assuming that it can be extended out to more hours by simply multiplying the hourly rate by how many hours they work. For example, in our above example, our bettor was working 211.5 hours a month and was making $14 an hour. Their profit for the month was $2961 ($14*211.5 hours).

So, what happens if you work 212.5 hours? Do you make ($2961+$14) for your one additional hour of work? The answer is maybe, but probably not. If you research one more hour, your picks may get slightly better, and you make more money. Or it’s completely possible that they don’t change at all and you make the same amount of profit, and your hourly rate thus goes down.

There may be diminishing returns with some of the time investments you have with your form of betting. This is heavily dependent on where your additional time is being placed and on which form of betting you are working with. Let’s look at another example from a poker player that will help us to see a few more things that need to be visible here.

Let’s say we have a poker player who plays 20 hours a week and studies 10 hours a week and makes $500 a week. This player has a current win rate of $16.67 per hour ($500/30). If this player adds one more hour to their study time, it’s impossible to predict what will happen to their win rate. It may go up because they are learning more things, but it’s hard to say how much. It may stay the same for a while and then go up or stay the same forever. Or it could go down because the additional hour is overworking the player. The point is that it’s hard to say what that extra hour is going to do and you can’t just assume you will be making another $16.67.

What about if that player adds an additional hour at the tables? With this extra hour, it is much more likely that the player is going to increase their win rate somewhere close to the $16.67. However, this still may not be the case. It’s possible that this player is playing their 20 hours on Friday and Saturday nights when the games are really juicy and easy to beat. There may not be another hour of this good of an opportunity. Playing poker on a Friday night at 9 pm is going to be more lucrative than playing at 10 am on a Tuesday.

There may not even be any games to play on that Tuesday morning and the 20 hours may be the max available. This is the same with sports bettors. If you are already betting all of the games for a particular sport, you can’t increase your win rate anymore by “betting for more hours.”

We could give you examples and hypotheticals all day but here is the bottom line. You cannot assume that each hour or hours you add to your time will increase your profit at the same hourly rate. Why is this important? Let’s say you are dedicating 20 hours to sports betting and making $500 a week and trying to decide if you should quit your normal job to go full time. You can’t just assume that if you work 40 hours you are going to make $1000 a week.

People make this assumption ALL the time. If you account for all the variables, it can help you to have an idea of the potential, but in reality, you are going to need a much more complex formula or some trial and error to see what happens. Try putting in 21 hours and see what happens to your bottom line. Try 22 hours, try 23, etc. You just need to make sure that you don’t make a rash decision without realizing that these win rate statistics are not usually able to be linearly extrapolated.

Variance Concerns

People that are new to betting have a tendency to view win rates as guaranteed payment. They’re usually used to getting an hourly paycheck or a paycheck for a fixed amount and enjoy the comfort in that. When they calculate their win rate, they assume that they are going to be getting that amount no matter what and plan their life accordingly.

The problem with this is that your win rate is not guaranteed. The gambling fairy does not deliver you a paycheck every month that says you get x amount for working y amount of hours. You have to continue to perform, and there will be variance in your win rate. Yes, the longer the period of time your stats are from the more accurate the win rate is going to be. The point is that there will be weeks and months where you make less than your win rate, and of course, there will be some weeks and months where you make more.

Most of this will balance out (thanks to the law of large numbers) if you have a strong win rate calculation, but thanks to variance, you may not get the exact amount you’re counting on every month. Things may also change in the industry, and your win rate may be going through a change as well that you may not be aware of. The point is this: If your win rate is $10, don’t count on getting $400 if you work 40 hours the next week. Yes, you may get this, but it’s entirely possible that you could get less or more. If you set your life up where you are counting on that money (living paycheck to paycheck), you’re never going to make it.

Constantly Updating

An issue we see a lot is that players and bettors will calculate their win rate once and then use that number for months or years at a time. The problem with that is that the numbers are always changing and you need to be constantly updating your win rate. The game may be getting harder or easier, you may be getting better or worse, expenses may be increasing or decreasing, or there may be a whole host of other variables that will change your calculations.

The best solution is to keep a spreadsheet that is constantly updating. Always track and update your expenses as well as your wins, losses and time put into it. This will ensure that you are always aware of what your win rate is and if it is staying constant, increasing, or (hopefully not) moving downward in the wrong direction. Keeping great records and having data you can actually work with is the key to longevity.