Suppose you are at a casino and wish to play only one game the entire night that will give you the best chance of winning. Should you play the slots, hit the blackjack table, roulette, craps or something else? I cannot calculate the probability of winning the other games in the scope of an article this size. But I can tell you that craps is your best option. But beware, even with craps there is a better chance you go home a loser than a winner. I’ll explain why.
Rules of the game
A player throws two six-sided dice. If the sum is 7 or 11, then he wins. If the sum is 2,3 or 12, he loses. If the sum is anything else, he continues throwing until he either throws that number again (in which case he wins) or throws a 7 (in which case he loses).
Probability of winning
To find the probability of winning, first find the probability of winning on the first roll by getting a 7 or 11. There are 6 ways to get a sum of 7 on two dice and the number of possible outcomes when rolling two dice is 36. Therefore, the probability of getting a 7 is 1/6. There are 2 ways to get a sum of 11 on two dice. Therefore, the probabiitly of getting 11 is 2/36 or 1/18. Adding these two probabilities gives the probability of winning on the first roll, which is 2/9.
Next, find the probability of losing on the first roll, which is getting the sum of 2,3 or 12 on two dice. There is only one way to get the sum of 2 and one way to get a sum of 12 on a pair of dice. There are two ways to get a sum of 3 on a pair of dice. Adding the probabilities we get 1/36 + 1/36 + 1/18 = 1/9.
You can continue playing by rolling a sum of 4,5,6,8,9, or 10. The probability of such is 1 – 2/9 – 1/9 = 2/3. You simply subtract the probabilities of getting 2,3,7,11, and 12 that we obtained above, from 1.
Using more complex probability, we can figure out the probability of rolling a 4 and eventually winning to be 1/36, which is the same as the probability of rolling a 10 and eventually winning since there is equal chance of rolling a sum of 4 and a sum of 10. The probability of rolling a sum of 5 or 9 and eventually winning is 2/45. Finally, the probability of rolling a sum of 6 or 8 and winning is 25/396.
Putting it all together: Final analysis
After calculating the probabilities, we get the probability of winning equals the probability of winning on first roll + probability of winning rolling a 4 or 10 first + probability of winning rolling 5 or 9 first + probability of winning rolling 6 or 8 first. This equals 0.493. Therefore you have a 49.3% chance of winning at craps assuming you are playing with two fair six-sided dice.
If you want to just have fun, play the games you enjoy the most. After all, even the game with the best chance of winning still favors the casino by a 50.7% to 49.3% margin.
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