5. (Sudden death) The NHL has another season-long strike, but the owners
and players reach an agreement in June which leaves time for a highly
abbreviated season. They decide that fans want to see the Stanley Cup
decided, and so they play only a sudden-death version of the seventh
game of the final round of the playoffs. Here are the rules:
In each round, Montreal and Boston each take two penalty shots. If
one team makes more goals than the other, it wins the Stanley Cup.
Otherwise another round is played. This continues until there is a winner.
All shots are independent, but Montreal has a probability 2/3 of scoring
a goal on an individual penalty shot, while Boston has a probability of
only 1/2 .
(a) Denote by P(M) the probability that Montreal wins the Cup. This
is also the conditional probability that they win the Cup, given that
the first round was inconclusive. Use this insight to set up and solve
an equation for P(M).
(b) Event A is “the Cup was decided on the first round;” event M is
“Montreal won the Cup.” Are these events independent? If not,
what is P(M|A)?