Probability in a Table Tennis Match

Let’s dive into a probability problem involving a table tennis match between Sakura and Emily. This is a great example of how probability concepts can be applied to real-world scenarios.

The Scenario:

Sakura and Emily are playing a table tennis match. The first player to win three games wins the match. Sakura’s probability of winning any single game is 0.55, and each game is independent of the others. Games cannot be drawn.

(i) Sakura Wins in Three Games:

This is the simplest scenario. For Sakura to win in three games, she needs to win all three games consecutively. Since the games are independent, we can multiply the probabilities:

P(Sakura wins in 3 games) = P(Sakura wins game 1) * P(Sakura wins game 2) * P(Sakura wins game 3)
= 0.55 * 0.55 * 0.55
= 0.55³
≈ 0.166

So, the probability that Sakura wins the match in three games is approximately 0.166.

(ii) Emily Wins the Match:

This is a bit more involved. Emily can win in three, four, or five games. It’s often easier to calculate the probability of Sakura winning the match and subtract it from 1. However, let’s calculate Emily’s winning probabilities directly.

• Emily wins in 3 games:P(Emily wins in 3 games) = 0.45³ ≈ 0.091
• Emily wins in 4 games:For Emily to win in 4 games, she must win the last game and two of the first three. Sakura wins one of the first three. So possible outcomes are SEEE, ESEE, EESE.
  P(Emily wins in 4 games) = 3 * (0.55 * 0.45 * 0.45 * 0.45) = 3 * 0.55 * 0.45³ ≈ 0.150
• Emily wins in 5 games:For Emily to win in 5 games, she must win the last game and two of the first four. Sakura wins two of the first four. The possible outcomes are SSEE, SESE, SEES, ESSE, ESES, EESS,
  Therefore Sakura can win two games in \binom{4}{2} ways. And Emily must win the 5th game.
• P(Emily wins in 5 games) = \binom{4}{2} * (0.55² * 0.45² * 0.45) = 6*0.55² * 0.45³ ≈ 0.183

Therefore, the probability that Emily wins the match is:

P(Emily wins) = P(Emily wins in 3 games) + P(Emily wins in 4 games) + P(Emily wins in 5 games)
≈ 0.091 + 0.150 + 0.183
≈ 0.424

So, the probability that Emily wins the match is approximately 0.424.

This type of problem is excellent for Year 13 A Level Maths Revision because it combines basic probability principles with a need for careful consideration of different scenarios. It’s important to break down the problem into smaller parts and consider all the possible ways each player can win.