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Data Interview Question

a 7-Game Series

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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(x wins)=(nx)px(1p)nxP(\text{x wins}) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}

    Where:

    • nn = Total number of games (6 in this case)
    • xx = Number of wins (3 wins for each team)
    • pp = 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:

    (63)=6!3!(63)!=6×5×43×2×1=20\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(3 wins for each team)=(63)(0.5)3(0.5)3P(\text{3 wins for each team}) = \binom{6}{3} \cdot (0.5)^3 \cdot (0.5)^3

    • Simplify:

    P(3 wins for each team)=20×(0.5)6=20×164=2064=516P(\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 516\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 516\frac{5}{16}?

    • The calculation shows that out of all possible outcomes of the first 6 games, 516\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.