Tossing the Coin

Probability Level 3

Wayne and Mario play a game in which they take turns flipping a fair coin. The first one to flip tails win. Wayne goes first. What is the probability that Wayne wins?

The answer will be of form A B \frac{A}{B} where A A are B B are positive co-prime integers. Find A + B + A × B A+B+A \times B .

Note: this problem is taken from AoPS


The answer is 11.

Arulx Z by Arulx Z , 20, India

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1 solution

展豪 張
Mar 7, 2016

A standard solution using GS:
Probability that Wayne wins in the 1st time is 1 2 \frac 12
that she wins in the 2nd time is ( 1 2 ) 3 (\frac 12)^3
...
The total probability is the sum:
1 2 + ( 1 2 ) 3 + . . . \frac 12 + (\frac 12)^3 + ...
= 1 2 1 1 − 1 4 =\frac 12 \frac 1{1- \frac 14}
= 2 3 =\frac 23


4 Helpful 0 Interesting 0 Brilliant 0 Confused

That's correct! Only possibilities for Wayne to win are T, HHT, HHHHT and so on

Arulx Z - 5 years, 3 months ago

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