Fermat solved the problem of points (see Exercise) as follows: He realized that

Fermat solved the problem of points (see Exercise) as
follows: He realized that the problem was difficult because the possible ways
the play might go are not equally likely. For example when the first player
needs two more games and the second needs three to win two possible ways the
series might go for the first player are WLW and LWLW. These sequences are not
equally likely.To avoid this difficulty Fermat extended the play adding fictitious
plays so that the series went the maximum number of games needed (four in this
case).He obtained equally likely outcomes and used in effect the
Pascal triangle to calculate P(r s). Show that this leads to a formula for
P(r s) even for the case p =1/2.