Question

Alice, Bob, and Charlie are rolling a fair die in that order. They keep rolling until one of them rolls a 6.

What is the probability that each of them wins?

Answer 1

a)The prob of Alice win = Alice wins in the first round + Alice Wins in the 2nd round + Alice Wins in the 3nd round+---

=1/6 + 5/6*5/6*5/6*1/6 + 5/6*5/6*5/6*5/6*5/6*5/6*1/6 +----- (1/6 is the probability of getting 6 and 5/6 is the probability of getting a number which is not 6 while rolling a fair dice)

Since you can see that the above is a GP with r<1

So we can use the equation a/(1-r) to get the ans

where a=1/6 and r = (5/6*5/6*5/6*1/6)/(1/6) = 125/216

So., Prob of Alice win = (1/6)/(1-125/216) =0.3956044

b)Same as above nowdoing for Bob but his turn comes after Alice so

The prob of Bob win = Bob wins in the first round + Bob Wins in the 2nd round + Bob Wins in the 3nd round+---

=5/6*1/6 + 5/6*5/6*5/6*5/6*1/6 + 5/6*5/6*5/6*5/6*5/6*5/6*5/6*1/6 +----- (1/6 is the probability of getting 6 and 5/6 is the probability of getting a number which is not 6 while rolling a fair dice)

Since you can see that the above is a GP with r<1

So we can use the equation a/(1-r) to get the ans

where a=5/6*1/6 and r = (5/6*5/6*5/6*5/6*1/6)/(5/6*1/6) = 125/216

So., Prob of Bob win = (5/6*1/6)/(1-125/216) =0.3296703

c)

Same as above nowdoing for Charlie but his turn comes after Alice&Bob so

The prob of Charlie win = Charlie wins in the first round + Charlie Wins in the 2nd round + Charlie Wins in the 3nd round+---

=5/6*5/6*1/6 + 5/6*5/6*5/6*5/6*5/6*1/6 + 5/6*5/6*5/6*5/6*5/6*5/6*5/6*5/6*1/6 +----- (1/6 is the probability of getting 6 and 5/6 is the probability of getting a number which is not 6 while rolling a fair dice)

Since you can see that the above is a GP with r<1

So we can use the equation a/(1-r) to get the ans

where a=5/6*5/6*1/6 and r = (5/6*5/6*5/6*5/6*5/6*1/6)/(5/6*5/6*1/6) = 125/216 (i.e. 2nd term / first term)

So., Prob of Charlie win = (5/6*5/6*1/6)/(1-125/216) =0.2747253

Hope the above answer has helped you in understanding the proble. Please upvote the ans if it has really helped you. Good Luck!!