A and B take turns rolling a die. A starts, the winner is the first one that rolls a 6. What is the probability that A wins?

Guest May 14, 2015

#1**+15 **

Probability that A wins on first throw = 1/6

Probability that A and B lose their first throws and A wins on his 2nd throw = (5/6)^{2}1/6

Probability that A and B lose their first two throws and A wins on his 3rd throw = (5/6)^{4}1/6

etc.

So probability that A wins = (1/6)(1 + (5/6)^{2} + (5/6)^{4} + ...)

This is a geometric progression with sum = (1/6)/(1 - (5/6)^{2}) = 6/11

So the probability that A wins = 6/11

.

Alan May 14, 2015

#1**+15 **

Best Answer

Probability that A wins on first throw = 1/6

Probability that A and B lose their first throws and A wins on his 2nd throw = (5/6)^{2}1/6

Probability that A and B lose their first two throws and A wins on his 3rd throw = (5/6)^{4}1/6

etc.

So probability that A wins = (1/6)(1 + (5/6)^{2} + (5/6)^{4} + ...)

This is a geometric progression with sum = (1/6)/(1 - (5/6)^{2}) = 6/11

So the probability that A wins = 6/11

.

Alan May 14, 2015