Professional sports leagues commonly end their seasons with...

Professional sports leagues commonly end their seasons with a championship series between two teams. The series ends when one team has won four games and so must last at least four games and at most seven games. How many different sequences of game winners are there in which the series ends in a. 4 games? b. 5 games? c. 6 games? d. 7 games? e. Assuming that the two teams are evenly matched, determine the probability of each of the outcomes in parts (a)-(d).

Professional sports leagues commonly end their seasons with a championship series between two teams. The series ends when one team has won four games and so must last at least four games and at most seven games. How many different sequences of game winners are there in which the series ends in
a. 4 games?
b. 5 games?
c. 6 games?
d. 7 games?
e. Assuming that the two teams are evenly matched, determine the probability of each of the outcomes in parts (a)-(d).

Key Concepts

Combinatorial Enumeration

This concept involves counting the number of ways that a series of events can occur under specific constraints. In the context of a championship series, it means counting the number of game sequences that lead to one team winning the series, taking into account that the series stops as soon as a team reaches four wins.

Binomial Coefficient

Binomial coefficients are used to calculate the number of ways to choose a subset of items (in this case, wins or losses) from a larger set. They are particularly important when determining how many sequences contain a fixed number of wins for the losing team before the decisive game.

Probability of Sequential Events

This involves calculating the likelihood of a particular sequence of outcomes in a series of independent events, such as the results of games in a sports series. When the teams are evenly matched, each game outcome is equally likely, and the overall probability can be computed by considering the probability of each individual game result in the sequence.

Conditional Stopping Rules

In many sequential processes, the event of interest (such as winning a championship) occurs only when a specific condition is met. For a championship series, the process stops once a team wins four games. This condition must be incorporated into the counting process to ensure that only valid sequences — those that stop immediately when the necessary win total is reached — are considered.

Transcript

00:01 So in this problem, we're dealing with a scenario with championship series and sports games.

00:06 And basically the premise is that two teams are assuming they're evenly matched, will play until one team wins four games.

00:13 So there will be a maximum of four games in the series, or a minimum of four games in the series, and a maximum of seven.

00:20 And in this problem, we're being asked to calculate the different number of ways that the teams can end a championship series in different numbers of games.

00:29 So in part a, we're being asked, how many different sequences of game winners are there in which a series ends in four games? and if a series ends in four games, that means one team wins all of the games.

00:44 And since there are only two teams, there are two ways because either one team wins all of the games or another team wins all of the games.

00:53 In part b, we are doing this calculation but for a five game series.

00:58 And if five games are played, that means one team loses one of the first four games and wins the other games, and wins the other three games, as well as the last game.

01:11 And so in this case, we're going to do a permutation, and the order of the games is important, so that is why we're doing a permutation.

01:20 And basically, we want to figure out how many ways there are for one team to lose one game in the first four games.

01:27 So to do this, we are choosing from the first four games, or we are doing a permutation, and they are losing one game.

01:37 So it's going to be four factorial over four minus one factorial, which is four factorial over three factorial, which is simply four.

01:51 And since there are two teams that could be the winning team, we need to multiply this number by two.

01:58 So 2 times 4 is equal to 8, and that means there are 8 different ways in which these 5 games could be played...

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