Past exam of the mathematics course of the University of Cambridge / 2001 / ia / Paper 2 / 9F / Solution

Let the win probabilities in games 1,2,3 be $u,v,w$. Independence makes a prescribed result pattern the product of its three win/loss probabilities. Taking a complement for event (a), inclusion-exclusion for overlapping pairs in (b) and (c), and disjoint exact-two patterns in (d) and (e), givesFor (b), a three-win outcome is counted three times in the pair sum and must be counted once, so subtract twice its probability. For (c), only the overlapping pairs 12 and 23 are allowed, so subtract the three-win intersection once. For (d), the two allowed exact patterns are 110 and 011. For (e), add 101 as well.

Substituting $(u,v,w)=(q,p,q)$ and $(p,q,p)$ gives the complete comparison:For (a) and (b), the differences $P_{MDM}-P_{DMD}$ are respectively $(p-q)(1-p)(1-q)>0$ and $(p-q)(p+q-2pq)>0$. For (c) and (d), the differences $P_{DMD}-P_{MDM}$ are $pq(p-q)>0$ and $2pq(p-q)>0$. For (e), $P_{MDM}-P_{DMD}=(p-q)(p+q-3pq)>0$ by the additional hypothesis. Thus all maximizing orders follow with strict inequalities.