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?
Probability of first player winning
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probability-distributions
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0What 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. – 2012-10-29
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0I dont see how they are equivalent, please elaborate. – 2012-10-31
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2Vaolter 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. – 2012-10-31
2 Answers
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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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1How is did you get this equation, I am confused. – 2012-10-31