The Use of Mathematics in Gambling

I thought it would be fun to include a page of gambling mathematics on the site.

One of the most rewarding and entertaining aspects of being a writer of casino and poker subjects is the chance to study and create practical math problems.

Most of the problems on this page are pretty basic.

I’m not smart enough to write a test with really hard problems on it, although I did get an almost-perfect score on the mathematics portion of my SAT. (I missed two questions.)

Betting on the Toss of a Coin

George has created a low-rent casino game based on flipping a coin. If the coin lands heads-up, the player wins the coin. If the coin lands tails-side up, the player loses the coin.

The player has to make an ante bet in order to play. This ante bet is 10% of the size of the original bet.

What is the house’s edge over the player in this situation?


Half the time the casino wins a unit; the other half the time the player wins a unit. 100% of the time, the casino wins 10% of a unit. This makes the casino’s edge in this game exactly 10%.

A Blackjack Problem

Michelle is playing blackjack. She has a hard total of 18, or a 10 and an 8.

What is the probability that she’ll bust if she takes another card?


There are 50 cards left in the deck. The only cards that will prevent her from busting are the 3s, the 2s, and the As, of which there are 4 each (12 total).

12 divided by 50 is 24%, which means that she’ll bust 76% of the time in this situation if she takes another card.

Choosing between Multiple Bets

You’ve found a casino where you can count cards with a 1% edge over the house. You also have the option of playing video poker with a 0.1% edge over the house. The blackjack game has a maximum bet of $20 per hand.

The video poker game has a maximum bet of $5 per hand, but you can play 52 hands at once. You have more than enough bankroll to play either game. You can play 300 hands of video poker per hour, or you can play 200 hands of blackjack per hour

Which game offers you the higher expected profit per hour?


First you calculate how much money you can get into action each hour. In the blackjack game, you multiply $20 by 200 hands per hour, for a total of $4000 in action per hour.

In the video poker game, you multiply $5 by 52 hands by 300 hands per hour, for a total of $78,000 in action per hour.

You then multiply the total amount of action per hour by the advantage you have. In the blackjack game, you expect to win 1% of $4000 per hour, or $40 per hour. In the video poker game, you expect to win 0.1% of $78,000 per hour, or $78 per hour.

The video poker game provides you with the higher hourly expected win rate.

A Poker Hand

You’re playing five card draw and you have four cards to a flush.

What are the odds of drawing the fifth card of that flush and making your hand?


Many gamblers assume that the odds of getting that additional card are 1 in 4, or 25%. That’s because there are suits, and the next card could conceivably be the additional card of that suit.

What they’re not taking into account is the fact that 4 cards of that suit aren’t in the deck any more. The actual calculation looks like this.

You have 47 cards left in the deck. 9 of those cards are of the same suit as you need to fill your flush. (13-9=4). 9 divided by 47 is about 19.14%, which is considerably less than 25%.

A Casino Game Based on Dungeons & Dragons

“Gary” has created a casino game based on Dungeons & Dragons. The game is played by rolling a single twenty-sided die (a “d20). If you roll an 11 or higher, you win even money on your bet. If you roll a 10 or less, you lose your bet.

The game has two wrinkles, though. A roll of 20 is a “critical hit”, and you get paid off at 3 to 2 for that bet.

A roll of 1 is a “fumble”. In that case, you have to put up another bet and re-roll. If you win this second bet, you only get paid on the amount of your original bet.

If you lose, then both bets are lost. You cannot get a critical or fumble on this second roll.

What is the house edge for this game?


In 9 cases out of 20, or 45% of the time, the player wins the bet. In another 9 cases out of 20, or another 45% of the time, the player loses the bet. In these instances, the house edge is 0%. Neither the player nor the casino has an edge.

In 1 case out of 20, or 5% of the time, the player wins 1.5 units. This adds 0.5 units multiplied by 5% to the player’s edge, or +0.025.

In 1 case out of 20, the player has to put up 2 units in order to win 1 unit. He’ll lose this bet half the time for a total loss of 2 units. When he does win (the other half the time), he’ll only win 1 unit. This adds 1 unit multiplied by 5% of the casino’s edge, or -0.05.

+0.025 – 0.05 = 2.5%, which is the casino’s edge over the player in this game.

Here’s another way to look at it. Most of the time, this is an even money game. But 5% of the time, the player has a chance of winning 1.5 to 1. Another 5% of the time, the casino has a chance of winning 2 to 1.


Gambling is all about mathematics. If you can’t solve math problems, you shouldn’t gamble, especially not in a casino. Gambling without understanding the mathematics behind the games can leave you at a severe disadvantage.

It’s like going to the store and not understanding what anything is worth, but shopping anyway. You’ll wind up spending more money than you should have.

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