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Design a game where three coins are tossed and the player "wins" if he or she tosses either all three heads or all three tails. The question is if the "wager" is $1, then what is a fair winning payout for the player such that he or she would break about even playing this game over time?

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The (homework) tag was added inappropriately: the OP has not acknowledged this as homework. chipsae, if it is homework, you should add the tag: that won’t keep people from answering. – Brian M. Scott Feb 28 '13 at 7:48
@Qiaochu: You don’t have a legitimate reason. The only legitimate reason is acknowledgement by the OP that the tag is warranted. – Brian M. Scott Feb 28 '13 at 7:52
@Brian: I don't see it as insulting. I just don't want to see homework questions on the main page anymore, have ignored the homework tag for this purpose, and am retagging this question so that I don't have to see it. Other users are free not to ignore the homework tag or even to favorite the homework tag if they enjoy helping people who post homework questions. I just don't happen to be one of those users. In general, if some users want to see questions that should belong to a certain tag and other users don't, both groups of users benefit from... – Qiaochu Yuan Feb 28 '13 at 7:55
@Brian: I don't know a lot of things. For example, I don't know whether the Sun will rise tomorrow. I suspect, with considerable justification, that it will, so I continue to act as though it will. I don't see anything unreasonable about this. – Qiaochu Yuan Feb 28 '13 at 8:02
@QiaochuYuan, when you wrote "users other than the OP are allowed to edit tags ... I see no reason that the homework tag should be an exception" you must have been aware of the overwhelming consensus on meta, including the voiced agreement of users who dislike homework being posted, that the homework tag not be applied without explicit indication of HW by the OP. If you were some random user I suppose it would not matter as much. Having been elected as moderator, don't you feel that the often expressed consensus trumps your personal convenience in ignoring tags? – zyx Mar 10 '13 at 23:30

HINT: Only two of the possible outcomes, TTT and HHH, are wins. The other $n$ are losses, where I leave it to you to determine just what $n$ is. The $n+2$ outcomes are equally likely. If the winning payout is $p$, therefore, on average the player pays $n+2$ dollars and wins $2p$ dollars per $n+2$ games and has net winnings of $2p-(n+2)$ dollars. In a fair game the net winnings are $0$.

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The probability of $3$ heads in a row is $\left(\frac{1}{2}\right)^3$, as is the probability of $3$ tails in a row. So the probability of winning is $\frac{1}{4}$.

It is not precisely clear what "payout" means. We assume that you get your dollar back plus an amount $a$ that we will call the payout,

Then the random variable $X$, which is the amount you win, is $-1$ with probability $\frac{3}{4}$ and $a$ with probability $\frac{1}{4}$.

It follows that $E(X)=(-1)\left(\frac{3}{4}\right)+(a)\left(\frac{1}{4}\right)$. For a fair game, we need $E(X)=0$. Solve the equation $(-1)\left(\frac{3}{4}\right)+(a)\left(\frac{1}{4}\right) =0$ for $a$. We get $a=3$.

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