# Thread: Fun extra credit question

1. ## Fun extra credit question

Mary and Paul are playing a game of tossing a coin. Mary says to Paul, "We will toss the coin until the first head appears. If the first head appears after an odd number of tosses, I win the game. If the first head appears after an even number of tosses, you win the game." They each put $60 into a cup and when one of them wins a game, he or she removes a dollar from the cup. They continue to do this until the cup is empty. Mathematically speaking, how much money will each person remove from the cup? 2. Originally Posted by amiv4 Mary and Paul are playing a game of tossing a coin. Mary says to Paul, "We will toss the coin until the first head appears. If the first head appears after an odd number of tosses, I win the game. If the first head appears after an even number of tosses, you win the game." They each put$60 into a cup and when one of them wins a game, he or she removes a dollar from the cup. They continue to do this until the cup is empty. Mathematically speaking, how much money will each person remove from the cup?
Pr(Head on odd toss) = $\frac{1}{2} + \left( \frac{1}{2} \right)^3 + \left( \frac{1}{2} \right)^5 + \, .... = x$

Pr(Head on even toss) = $\left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^4 + \left( \frac{1}{2} \right)^6 + \, .... = \frac{1}{2} \left[ \frac{1}{2} + \left( \frac{1}{2} \right)^3 + \left( \frac{1}{2} \right)^5 + \, .... \right] = \frac{x}{2}$.

Therefore $x + \frac{x}{2} = 1 \Rightarrow x = \frac{2}{3}$.

So out of 60 games, how many is Paul expected to win ....?