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In a dice game craps, Alex rolls a pair of fair dice.

  • If he gets $7$ on the first roll, he wins immediately
  • If the result is any number $\neq 7$, he keeps rolling the dice until he gets that number again (he wins) or he get a $7$ (he loses).

Example: if Alex gets $8$ on the first roll, he keeps rolling the dice. After a number of rolls, if he gets $8$, he wins. If he gets $7$, he loses. I think of a probability but it seems infinite cases. I'm thinking of a conditional probability.

  1. What is the probability that Alex will win this game?
  2. What is the probability that Alex's final roll will be $7$?
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1 Answer 1

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The probability for a $7$ is $\frac 16$, the probability for $x\ne 7$ varies from $\frac1{36}$ to $\frac5{36}$. For example, when trying to get $8$ (probability $\frac{5}{36}$), the probability of succeeding before getting a $7$ is $\frac{5}{5+6}$ as the $36-5-6$ rolls giving neither $7$ not $8$ can be ignored. The probability of first rolling $8$ and then winning is therefore $\frac5{36}\cdot \frac{5}{5+6}$. By adding, the total probability of winning is $$\frac1{36}\cdot \frac{1}{1+6}+\frac2{36}\cdot \frac{2}{2+6}+\frac3{36}\cdot \frac{3}{3+6}+\frac4{36}\cdot \frac{4}{4+6}+\frac5{36}\cdot \frac{5}{5+6}+\frac6{36}+\frac5{36}\cdot \frac{5}{5+6}+\frac4{36}\cdot \frac{4}{4+6}+\frac3{36}\cdot \frac{3}{3+6}+\frac2{36}\cdot \frac{2}{2+6}+\frac1{36}\cdot \frac{1}{1+6} = \frac{6557}{13860}.$$ The probability that the game end by rolling a $7$, is $$1- \frac{6557}{13860}+\frac16=\frac{9613}{13860}. $$

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  • $\begingroup$ thanks for your response. this craps problem has small question. let X equal the number of times Alex wins and Y equal the number of times Alex's final roll is 7. suppose Alex plays 100 games (I guess what it means is that Alex rolls the pair of dice 100 times). what type of random varibles are X and Y? X and Y are independent? find their means and Standard deviation? How X and Y are correlated? $\endgroup$ Commented May 10, 2013 at 10:09
  • $\begingroup$ I think X and Y are discrete random variables. but not sure how to derive its means and standard deviation $\endgroup$ Commented May 10, 2013 at 10:10
  • $\begingroup$ I got all. thanks $\endgroup$ Commented May 10, 2013 at 12:08

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