a 7-Game Series

Solution & Explanation

To determine the probability that a best-of-7 series between two evenly matched teams goes to a decisive 7th game, we need to consider the conditions required for the series to reach this point. Specifically, both teams must win exactly 3 games each in the first 6 games.

Step-by-Step Solution:

1. Understanding the Problem:

   • You have two teams, A and B, each with a 50% chance of winning any individual game.
   • A best-of-7 series means that the first team to win 4 games wins the series.
   • We want to calculate the probability that the series is tied 3-3 after 6 games, necessitating a 7th game.

2. Mathematical Representation:

   • The problem can be modeled using a binomial distribution.
   • We need to calculate the probability of both teams winning exactly 3 games each in the first 6 games.

3. Binomial Probability Formula:

   The probability of a specific sequence of wins can be calculated using the binomial probability formula:

   $P(\text{x wins}) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}$

   Where:

   • $n$ = Total number of games (6 in this case)
   • $x$ = Number of wins (3 wins for each team)
   • $p$ = Probability of winning a single game (0.5)

4. Calculate Combinations:

   • Calculate the number of ways to distribute 3 wins to one team in 6 games:

   $\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$

5. Calculate Probability:

   • Using the binomial formula, calculate the probability:

   $P(\text{3 wins for each team}) = \binom{6}{3} \cdot (0.5)^3 \cdot (0.5)^3$

   • Simplify:

   $P(\text{3 wins for each team}) = 20 \times (0.5)^6 = 20 \times \frac{1}{64} = \frac{20}{64} = \frac{5}{16}$

6. Final Result:

   • The probability that the series goes to a decisive 7th game is $\frac{5}{16}$ or 31.25%.

Explanation:

• Why Binomial Distribution?

  • The binomial distribution is appropriate because each game is an independent trial with two possible outcomes (win or lose), and we are interested in the number of successes (wins) in a fixed number of trials (games).

• Why $\frac{5}{16}$?

  • The calculation shows that out of all possible outcomes of the first 6 games, $\frac{5}{16}$ of them result in both teams having won 3 games each, thus requiring a 7th game to determine the overall winner.

This approach provides a comprehensive understanding of how to calculate the probability of a 7-game series in a competitive environment with evenly matched teams.