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Two players take turns to toss a coin; the winner is the first to toss a head. What is the probability that the first player to toss the coin wins?

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What have you tried? This question basically amount to answer the question "What is the probability that when you flip a coin, the first head will appear on the first or third or fifth (and so on) flip?" This in turn amounts to summing an infinite series. –  JavaMan Oct 29 '12 at 18:03
    
I dont see how they are equivalent, please elaborate. –  Vaolter Oct 31 '12 at 11:06
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Vaolter Reading your reactions to the comment above and to an answer below, one cannot help to think that you might have explained what you tried and what you know about this problem, as they say one should do when asking a question here. –  Did Oct 31 '12 at 18:40
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2 Answers

up vote 1 down vote accepted

The probability that the first time a head is flipped is on the $i$th turn is $(1/2)^i$. Thus, the probability that the first person wins is

\begin{align} (1/2) + (1/2)^3 + (1/2)^5 + \dots \end{align}

Can you recognize this as a geometric series?

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Let's call $p$ the probability the first player wins

Assume first player tosses heads the first turn then he wins. If that is not the case then he tosses tails, hence the probability he wins is if the second loses which is equal to $1-p$ (the roles have been reversed)

Put it all in an equation $p= \frac{1}{2} \cdot 1 +\frac{1}{2} \cdot (1-p) \Rightarrow p=\frac{2}{3}$

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How is did you get this equation, I am confused. –  Vaolter Oct 31 '12 at 11:05
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