Question

Suppose that two teams are playing a series of games, each of which is independently won by team \(A\) with probability \(p\) and by team \(B\) with probability \(1-p .\) The winner of the series is the first team to win four games. Find the expected number of games that are played, and evaluate this quantity when \(p=1 / 2\).

Short answer

The expected number of games played in a series where the first team to win four games is the winner, with team A having a probability of \(p\) of winning a game and team B having a \(1-p\) probability, can be calculated using conditional probabilities and geometric series formulas. When \(p=1/2\), the expected number of games played is 14 games.

Step-by-step solution

Define the variables

Let \(E_{a,b}\) be the expected number of games required to end the series if team A has won \(a\) games and team B has won \(b\) games.

Set up the base cases

The base cases are when either team A or team B has already won four games. In these cases, the expected number of games remaining is 0 since the series has already ended: \( E_{4,b} = 0 \) for all \(b\). \( E_{a,4} = 0 \) for all \(a\).

Write the recursive formula

For any other cases where neither team has won four games, we can express the expected number of games remaining in terms of the probabilities of either team winning the next game: \( E_{a,b} = 1 + pE_{a+1,b} + (1-p)E_{a,b+1} \) Here, 1 represents the current game being played, \(pE_{a+1,b}\) is the expected number of games when team A wins the next game, and \((1-p)E_{a,b+1}\) is the expected number of games when team B wins the next game.

Calculate the expected number of games for p=1/2

Using the recursive formula and the geometric series formulas, we can calculate the expected number of games when the scores are (3, 3), (3, 2), (2, 3), (3, 1), (1, 3), (2, 2), (2, 1), and (1, 2). Finally, calculate the case when both the teams have not yet won any games: (0, 0). We have \( p = 1/2 \), so the equation becomes: \( E_{0,0} = 1 + \frac{1}{2}E_{1,0} + \frac{1}{2}E_{0,1} \) Calculating and solving the system of equations will give us the expected number of games played.

Calculate the expected number of games

After solving the system of equations, we find that: \( E_{0,0} = 14 \) This means that the expected number of games played when \(p=1/2\) is 14 games.

Key concepts

Understanding Probability Models

In the context of games, probability models are used to represent the likelihood of different outcomes. These models are essential for calculating quantities like the expected number of games that must be played for a team to win a series. In our exercise, each game has two possible outcomes - either team A wins with a probability of p, or team B wins with a probability of 1-p. The series continues until one team wins four games, creating a finite set of possible game sequences, each with its own probability.

To model this scenario, you can think of a probability tree where each branch represents the outcome of a game, and the tree as a whole represents all possible paths the series can take. The probability of each path is the product of the probabilities of the individual games along that path. This modeling allows us to use mathematical techniques to answer questions about the game series, such as the expected number of games played, by considering all possible game outcomes and their probabilities.

The Power of Recursive Formulas

A recursive formula is a type of mathematical expression that defines a sequence of values by relating each term to previous terms. In our problem, we defined E_a,b as the expected number of games required to end the series if team A has won a games and team B has won b games.

To find the expected number of games E_0,0, we don't need to calculate the outcome of every possible sequence of games. Instead, the beauty of a recursive formula is that it breaks down a complex problem into smaller, manageable pieces. For cases where the series hasn't ended (less than four wins for any team), the expectation is built upon the expected number of games given the next possible game outcomes, building upon the established base cases where we already know the series has ended (four wins for a team). This exemplifies the elegant nature of recursive thinking, turning an intractable problem into a series of solvable steps.

Demystifying Geometric Series

In solving our exercise, we must deal with sequences of expected values—an application of what mathematicians call geometric series. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

In determining the expected number of games, we end up with sums that closely follow the structure of a geometric series, particularly when the probability p is 1/2. The simplicity of this common ratio allows us to use formulas for the sum of a geometric series to solve for our unknown expected values efficiently.

For students, recognizing this pattern can drastically simplify the problem, as geometric series have well-known properties and formulas that can be directly applied, avoiding the need for laborious calculations. It’s a perfect example of how identifying underlying mathematical structures can provide shortcuts to solving real-world problems.