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We have the following problem:

Father of a tennis player tells her that he'll give her a prize if she wins two consecutive matches out of 3 played alternately between a club master and her father. Who should she choose to play first, if the probability of winning a match with club master is smaller?

Here's my intuition, let's mark a father by $F$ and club master by $M$. So we can choose one of those 2 match sequences: $FMF, MFM$. It looks like we should win more matches overall if we choose $FMF$, and we win a prize by winning either a match 1 and 2, or match 2 and 3 in both cases, choosing $FMF$ seems like a good idea...

So let's solve the problem. Let's mark probability of winning with a father by $f$, and with club master $m$. Assume the result of each match is independent. Let's count probability of winning a prize in both cases.

Case 1 - $FMF$, the sequences of match results that give us the prize are $wwl, lww, www$ ($w$-win, $l$-lose). Probability of each sequence is, in the same order $fm(1-f),(1-f)mf,fmf$, summing them all we get probability $2fm - f^2m=fm(2-f)$.

Case 2 - $MFM$, by similair logic we get probability $2fm-fm^2=fm(2-m)$.

$m<f$, so $2-f<2-m$, so $fm(2-f)<fm(2-m)$

So it seems that we should choose to play with club master first... I'm really confused here - this seems really counterintuitive to me. Could someone give me some intuition on what has just happened?

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    $\begingroup$ The result actually seems somewhat intuitive for me: in order for her to get a prize, she has to beat the father and the club master at least once each. If she chooses $FMF$, then she only has one chance to beat the club master. If she chooses $MFM$, then she just has to beat the club master one of the two times, and beat the father. $\endgroup$ – Marcus M Dec 6 '15 at 21:54
  • $\begingroup$ Why is it counterintuitive? If the middle one is $M$ and she loses that she has no chance of winning the prize. At least leaving the father in the middle she gets two chances of winning. $\endgroup$ – AnalysisStudent0414 Dec 6 '15 at 21:54
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My intuition concerning this problem is this: You want two chances against the club master, because that increases your chances of winning.

What if you replace the father with a, say, baby B. Then you can either choose to play BMB or MBM. You are hopeful that you can defeat the baby, but the club master is another matter.

You should thus choose the matches MBM, as the middle match is pretty easy compared to the two outer matches, which you therefore benefit from having two chances at winning.

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