## Friday, June 8, 2012

 `Suppose you are at a casino and want to know which game has the ` `highest probability of winning. It's actually the dice game Craps, which ` `is played as follows. ` ` ` ```The player throws two dice. If the sum is 7 or 11, then he wins. If the sum is 2, 3 or 12, then he loses. If the sum is anything else, then he continues throwing until he either throws that number again (in which case he wins) or he throws a 7 (in which case he loses). To find the probability of winning, first finding the ``` `probability of winning on the first roll by getting a 7 or 11. ` ` ` `The number of ways to get a sum of 7 on two dice is` ` ` `(1,6), (6,1), (2,5), (5,2), (3,4), (4,3) which is 6.` ` ` `The number of possible outcomes when rolling two dice is 36. ` `Therefore the probability of getting 7 is 1/6.` ` ` `There are 2 ways in which get get a sum of 11, (6,5) and (5,6).` `Therefore the probability of getting 11 is 2/36 = 1/18.` `  ` ```The probability of winning on first roll = 1/6 + 1/18 = 2/9 Next we 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 1 way to get a sum of 2 on a pair of dice, (1,1) so the probability` `of getting a 2 is 1/36.` ` ` `There are 2 ways to get a sum of 3 on a pair of dice (1,2), (2,1), so the ` `probability of getting a 3 is 1/18.` ` ` `There is only 1 way to get a sum of 12 on a pair of dice (6,6), so the probabiliy` `of getting a 12 is 1/36. ` ` ` `Therefore, the probability of losing on the first roll is 1/36 + 1/36 + 1/18 = 1/9.` ` ` `The probability of rolling a sum of 4,5,6,8,9 or 10 which enables you to continue` `rolling until you either win or lose is 1 - 2/9 - 1/9 = 2/3.` ` ` `You can also list the probability of obtaining each sum and adding, but there` `is no need to do this. We simply subtract the probabilities of obtaining all` `other possible sums (2,3,7,11,12) from 1.` ` ` `Now focus on the individual probabilities depending on which sum has ` `been obtained which enables you to keep rolling until you win or lose.` `The probability of getting 4 as is = 1/12 since there are 3 ways (1,3), (3,1)` `(2,2) of getting a 4 and 36 possible outcomes.` ` ` `The probability of getting 7 = 1/6` ```So the probability that the game continues = 1 - 1/12 - 1/6 = 3/4 So the probability of winning in this case is (1/12) + (3/4)(1/12) + (3/4)^2(1/12) + ..... = (1/12)[1 + (3/4) + (3/4)^2 + ..... to infinity] (factor out 1/12) = (1/12)[1/(1 - 3/4)] = (1/12)(4) = 1/3 Therefore the probability of getting a 4 and then eventually winning is = (1/12) (1/3) = 1/36.``` ` ` `Notice that this is the same as the probability of winning when rolling` `a 10 first because getting a 10 is equally likely as getting a 4 (3 possible ` `outcomes, (4,6), (6,4), (5,5).` ` ` ```Similarly the probability of getting a 5 or 9 is the same and is 1/9 since there are 4 possible ways to get a 5 and 4 possible ways to get``` `a 9 ( (1,4), (4,1), (2,3), (3,2) and (4,5), (5,4), (3,6), (6,3), respectively))` ` ` `The probability of continuing after is 1 - 1/9 - 1/6 = 13/18. ` ` ` ` ` ```Using the same reasoning as above the probability of winning is (1/9)[1/(1 - 13/18)] = (1/9)(18/5) = 2/5.``` ` ` ```The probability of getting a 5 or 9 and eventually winning is is (1/9)(2/5) = 2/45 ``` ` ` ` ` `The probability of getting a sum of 6 or 8` `is 5/36 ( (1,5), (5,1), (2,4), (4,2), (3,3) and (2,6), (6,2), (3,5), (5,3),` `(4,4) respectively ) ` ` ` `The probability of continuing after is 1 - 5/36 - 1/6 = 25/36` `The probability of winning is ` `(5/36)[1/(1 - 25/36)] = (5/36)(36/11) = 5/11 ` ` ` ```The probability of getting a 6 or 8 and eventually winning is (5/36)(5/11)= 25/396 ``` ` ` ```Since we went through all the possibilities, the total probability of winning = 2/9 + 2[1/36 + 2/45 + 25/396] ``` ` ` ` (probability of winning on 1st roll + probability of winning rolling 4 or 10` `first + probability of winning rolling 5 or 9 first + probability of winning` `rolling 6 or 8 first)` ` ` ` = 0.493` ` ` `So even the game with the best chance of winning, you will lose slightly more often` `than not. `