To win a prize you must win two games of chess in a row out of three, alternating between a strong player...

probability/two.wins.in.a.row New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

To win a prize you must win two games of chess in a row out of three,
alternating between a strong player and a weak one; should you play
strong, weak, strong or weak, strong, weak?

   I have to win two games in a row at chess to win a prize. I have to
   play only three games altogether, alternating between a strong player
   and a weak player. Should I play; strong, weak, strong
                                   or; weak, strong, weak ?

-- Philip Carter <philip@knowl.demon.co.uk>

[ Editor: the title sentence above is a summary; the original question
follows it. ]

Show solution

Play s,w,s

Reason:

Suppose your chance of beating the strong player is a,
and your chance of beating the weak player is b.

Obviously b>a>0

if play s,w,s, your chance of winning is

P(winning game 1,2 or win game 2,3)
        =P(win game 1,2)+P(win 2,3)-P(win 1,2 and win 2,3)
        =a*b+b*a-a*b*a    (1)


if play w,s,w, your chance of winning is

P(winning game 1,2 or win game 2,3)
        =P(win game 1,2)+P(win 2,3)-P(win 1,2 and win 2,3)
        =a*b+b*a-b*a*b     (2)


 b*a*b>a*b*a ==> (1) > (2)

-- Yidong <ding@fas.harvard.edu>

[ Editor: the point is that you must win the middle game, so it should
be the easier one. ]