Question

In his Letter to a Friend on Sets in Court Tennis, written in 1687 but not published until 1713, Jakob Bernoulli analyzed the probabilities at any point in a game or set of court tennis, whose scoring rules are virtually identical with those of tennis today. He determined the odds both when the players were evenly matched and when one player was stronger than the other. If two players \(A\) and \(B\) are evenly matched in a tennis game with the score \(15: 30\), determine the probability of player \(A\) winning. (Remember that one must win by two points.)

Short answer

Answer: The probability of player A winning the game is \(\frac{3}{16}\).

Step-by-step solution

Determine the probability of winning a point

Since both players are evenly matched, the probability of player \(A\) winning a point is equal to the probability of player \(B\) winning a point. Therefore, the probability of winning a point is: $$ p = \frac{1}{2} $$

Calculate the probability of \(A\) winning the game

We now need to consider the cases when \(A\) wins by two points or more, given the current score (15:30): 1. A wins the next 3 points: Probability = \((\frac{1}{2})^3 = \frac{1}{8}\) 2. A wins 2 points, and B wins 1 point before A wins another point: Probability = \((\frac{1}{2})^2 \times (\frac{1}{2}) \times (\frac{1}{2})^2 = \frac{1}{8}\) 3. A wins 1 point, and B wins 2 points (to get a deuce) before A scores another 2 points: Probability = \((\frac{1}{2}) \times (\frac{1}{2})^2 \times (\frac{1}{2})^2 = \frac{1}{16}\) Since these are the possible scenarios in which \(A\) wins the game, we add their probabilities together to get the overall probability of \(A\) winning: $$ P(A) = \frac{1}{8} + \frac{1}{8} + \frac{1}{16} = \frac{3}{16} $$

Present the result

The probability of player \(A\) winning a tennis game with the score \(15 : 30\) when both players are evenly matched is \(\frac{3}{16}\).

Key concepts

Jakob Bernoulli

Jakob Bernoulli was a Swiss mathematician born in the 17th century, and a prominent member of the Bernoulli family—a dynasty of mathematicians that made substantial contributions to the field. Jakob is best known for his work on probability theory and is often credited as one of the pioneers in the field. His work laid the foundation for the law of large numbers in probability theory.

One of Bernoulli's significant contributions to probability was through his book 'Ars Conjectandi' where he applied mathematical probability to games of chance, including the famous problem of points. His exploration into these games signified one of the earliest formal uses of probability theory and stoked future interest in the statistical and predictive potential of the science. The example of court tennis, as given in the exercise, reflects Bernoulli's profound interest in not just theoretical mathematics, but in applying mathematical principles to real-world scenarios like sports or gambling.

Mathematical Probability

Mathematical probability is a measure of the likelihood of an event occurring. It is quantified as a number between 0 and 1, with 0 indicating that an event is impossible and 1 that an event is certain to happen. The probability of all possible outcomes in a particular scenario will sum to 1.

Probability is based on observations or theoretical calculations, and in the context of games, probability often involves combinatorics to calculate the total number of possible outcomes. Bernoulli's work shows us that mathematical probability can predict the likelihood of various outcomes in games of chance or skill, like tennis, by constructing formulas or calculations that consider all possible scenarios under a set of fixed rules. The exercise problem utilizes Bernoulli's insights by analyzing the chances of a player winning given their current score in a game.

Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of objects. It serves as the foundation of probability theory, as it provides the tools for enumerating the possible arrangements, thus allowing for the calculation of probabilities. In the context of games and probability, combinatorics helps in determining the possible ways an event can occur.

In the exercise presented, we see combinatorics at work in determining the number of ways player A can win the tennis match from a specific score. It examines the sequences of winning and losing points that lead to an ultimate win. The calculations rely on understanding permutations and combinations of game points, which are fundamental concepts in combinatorics. This field's methods are crucial for solving probability problems in games, as seen in calculating player A's chance of winning the tennis match.