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You toss 4 fair coins. Dealer pays out $1 for each heads. The catch is that he may make you re-flip any one coin. What is the expected value of this game? Please show work.

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I assumed that dealer would make one reflip one coin (heads) aside from possibility of TTTT. –  QRIUS2KNW Aug 8 '13 at 0:13
"Please show work"... Funny, I was going to suggest exactly this. –  Did Aug 8 '13 at 0:14
Before I play this game, it might be nice if the rules were clear. If I get $3$ heads, does that mean that the dealer can order me to flip these $3$ coins again? And if she doesn't like the result, can she insist I do it again? –  André Nicolas Aug 8 '13 at 0:37
@AndréNicolas If you get 3 heads the dealer may only make you flip one of these heads. The dealer can not make you reflip all 3 in this case –  QRIUS2KNW Aug 8 '13 at 0:39
That's what I get. We are assuming of course that the dealer tries to maximize her company's expected income. So she will not ask us out of niceness to reflip a coin that landed tail! –  André Nicolas Aug 8 '13 at 1:02
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1 Answer

Since you know already a method and answer, my approach would be:

  • Value of game without reflip: $$4 \times \frac12 = 2$$

  • Value of reflip if have heads: $$-1 \times \frac12 = -\frac12$$

  • Probability of having heads: $$1-\frac{1}{2^4}=\frac{15}{16}$$

  • Value of game: $$2 -\frac12 \times\frac{15}{16} = \frac{49}{32}.$$

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