Example 2 gave the probability distribution for the number of games played in a best out of seven series between teams A and B. Let’s derive the probability distribution for a best of five series, in which the team that gets 3 wins first wins the entire series.
a. Write out the sample space of all possible sequences of wins and losses. (There are 20 such sequences. Constructing a [partial] tree diagram and stopping as soon as either 3 wins or 3 losses are observed may help in finding all 20.)
b. For each sequence in part a, determine its length. These are the distinct values of the random variable X = number of games played.
c. Find the probability of each sequence in part a when team A has a 50% chance of winning each game.
d. For each value x of the random variable X, find P(x) by adding up the probabilities for those sequences that end after x games.