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?