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# 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?

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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?

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)21/6

Probability that A and B lose their first two throws and A wins on his 3rd throw = (5/6)41/6

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

.

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)21/6

Probability that A and B lose their first two throws and A wins on his 3rd throw = (5/6)41/6

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
#2
+5

wow, thank you so much Alan, you just saved me.

May 14, 2015
#3
0

I like this solution too Alan, it is really neat May 15, 2015