Problem 10 - A History of Mathematics: An Introduction

Question

Continuing from Exercise 9, suppose that player \(A\) is twice as strong as player \(B\). Suppose that the score is \(30: 30\). Determine the probability of player \(A\) winning. What is the probability of \(A\) winning if the score is \(15: 30\) ?

Step-by-Step Solution

Answer

Answer: The probability of player A winning in both scenarios is \(\frac{4}{9}\).

Step 1: Scenario 1: Score 30:30 (Deuce)

In this scenario, we know that both players have scored 30 points each. To win, a player must score two consecutive points and therefore we can represent the winning probabilities as follows: - Probability of A winning first point: \(P(A1) = \frac{2}{3}\) (since A is twice as strong) - Probability of B winning first point: \(P(B1) = \frac{1}{3}\) - If A wins the first point, the probability of A winning the second point and the game: \(P(A2 | A1) = \frac{2}{3}\) Now let's calculate the probability of A winning the game using the conditional probabilities: \(P(A \ wins) = P(A1) * P(A2 | A1) = \frac{2}{3} * \frac{2}{3} = \frac{4}{9}\) Therefore, the probability of player A winning with a score of 30:30 is \(\boxed{\frac{4}{9}}\).

Step 2: Scenario 2: Score 15:30

In this scenario, player A has 15 points and player B has 30 points. Since B already won the first game, we only need to determine the probability of A winning two consecutive games to tie the score and eventually win. - Probability of A winning next game: \(P(A1) = \frac{2}{3}\) - Probability of A winning the game after that: \(P(A2) = \frac{2}{3}\) Now, let’s calculate the probability of A winning the match: \(P(A \ wins) = P(A1) * P(A2) = \frac{2}{3} * \frac{2}{3} = \frac{4}{9}\) Therefore, the probability of player A winning with a score of 15:30 is \(\boxed{\frac{4}{9}}\).

Key Concepts

Conditional Probability

Understanding conditional probability is essential when the outcome of an event is influenced by the occurrence of a previous event. In the given exercise scenario, when player A is at a score of 30:30, they must win two consecutive points to win the game.

The conditional probability, written as \(P(A2 | A1)\), denotes the probability of player A winning the second point given that they have already won the first point. This concept relies heavily on the dependency between the two events.

The formula for calculating conditional probability is:
\[ P(A2 | A1) = \frac{P(A1 \cap A2)}{P(A1)} \]
For player A, since winning the first and second points are independent events, the \(P(A1 \cap A2)\) (the probability that both events happen) is simply the product of the probabilities of each event occurring separately.

Probability Theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The exercise presents a practical aspect of probability theory—determining the likelihood of a player winning a game.

In theory, probability is always between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. For player A to win the game, two independent events must occur: winning the first point and then the second point. The multiplication rule for independent events states that:
\[ P(A \text{ wins}) = P(A1) \times P(A2) \]
Thus, probability theory provides the framework for calculating the odds of player A's victory in different score scenarios by considering the events' independence or dependence.

Mathematical Statistics

Mathematical statistics involves collecting, analyzing, interpreting, and presenting data. In the context of our exercise, we can relate it to making predictions based on observed data—such as player A's strength in comparison to player B.

The computation of player A's probability of winning from different scores can be used to infer patterns and make generalizations. For instance, despite different initial scores (30:30 versus 15:30), player A's probability of winning remained the same due to their consistent strength in the game.

This consistency aligns with the concepts of mathematical expectation and variance, which are fundamental to determining the predictability and variability of an event in statistics. The exercise demonstrates how a seemingly abstract concept plays a crucial role in predicting outcomes in real-world scenarios, such as games or sports competitions.