Question

Challenge Problem Tennis Anyone? Assume that the probability of winning a point on serve or return is treated as constant throughout the match. Further suppose that \(x\) is the probability that the better player in a match wins a set. (a) The probability \(P_{3}\) that the better player wins a best-of-three match is \(P_{3}(x)=x^{2}[1+2(1-x)]\) Suppose the probability that the better player wins a set is 0.6 . What is the probability that this player wins a best-of-three match? (b) The probability \(P_{5}\) that the better player wins a best-of-five match is $$ P_{5}(x)=x^{3}\left[1+3(1-x)+6(1-x)^{2}\right] $$

Short answer

Best-of-three: 0.648. Best-of-five: 0.89856.

Step-by-step solution

Understand the problem

The task is to determine the probability that the better player wins a best-of-three and a best-of-five tennis match, given that the probability of the better player winning a set is 0.6.

Use the given formula for best-of-three match

The formula to calculate the probability that the better player wins a best-of-three match is given by \[ P_{3}(x) = x^{2} \big[1 + 2(1-x)\big] \]where \( x \) is the probability that the better player wins a set. Here, \( x = 0.6 \).

Substitute and simplify for best-of-three match

Substitute \( x = 0.6 \) into the formula:\[ P_{3}(0.6) = (0.6)^{2} \big[1 + 2(1-0.6)\big] \]Calculate:\[ (0.6)^{2} = 0.36 \]\[ 1 + 2(1-0.6) = 1 + 2(0.4) = 1 + 0.8 = 1.8 \]Then,\[ P_{3}(0.6) = 0.36 \times 1.8 = 0.648 \]

Use the given formula for best-of-five match

The formula to calculate the probability that the better player wins a best-of-five match is given by \[ P_{5}(x) = x^{3} \big[1 + 3(1-x) + 6(1-x)^{2}\big] \]where \( x \) is the probability that the better player wins a set. Here, \( x = 0.6 \).

Substitute and simplify for best-of-five match

Substitute \( x = 0.6 \) into the formula:\[ P_{5}(0.6) = (0.6)^{3} \big[1 + 3(1-0.6) + 6(1-0.6)^{2}\big] \]Calculate:\[ (0.6)^{3} = 0.216 \]\[ 1 + 3(1-0.6) = 1 + 3(0.4) = 1 + 1.2 = 2.2 \]\[ 6(1-0.6)^{2} = 6(0.4)^{2} = 6 \times 0.16 = 0.96 \]Then,\[ 1 + 2.2 + 0.96 = 4.16 \]Finally,\[ P_{5}(0.6) = 0.216 \times 4.16 = 0.89856 \]

Key concepts

probability theory

Probability theory is a branch of mathematics that deals with the likelihood of events occurring.
It is fundamental in predicting outcomes in random processes, such as sports matches.
To understand a tennis match's outcome, we can use probability to estimate the chances of winning or losing.
For example, if the probability of a better player winning a set is given, we can calculate the chances of winning the entire match.
This is done using specific probability formulas.

algebra

Algebra helps us simplify and solve mathematical expressions using symbols and variables.
In our tennis problem, we use algebra to manipulate the given formulas to find the desired probabilities.
For instance, the formulas for calculating the probability of winning a best-of-three or best-of-five match, P_{3}(x) = x^{2} [1 + 2(1-x)] and P_{5}(x) = x^{3} [1 + 3(1-x) + 6(1-x)^{2}], involve algebraic operations like substitution and simplification.
These operations help us find the final probabilities.

sports mathematics

Sports mathematics applies mathematical principles to analyze and predict sports outcomes.
It can help coaches develop strategies, players improve their game, and fans better understand sports.
In tennis, we can calculate probabilities to find out how likely a player is to win matchpoints or sets.
For example, if a player's probability of winning a set is 0.6, we can determine the chances of winning a best-of-three match or a best-of-five match by plugging these probabilities into specific formulas.

best-of series

A best-of series is a competition where a certain number of games are played, and the first player to win the majority of those games wins the series.
In tennis, the best-of-three and best-of-five formats are common.
In a best-of-three match, the first player to win two sets wins. Similarly, in a best-of-five match, the first player to win three sets wins.
This format is crucial in calculating the overall probability of a player winning the series by analyzing each set's individual probabilities.

conditional probability

Conditional probability is the probability of an event occurring given that another event has already occurred.
It is a fundamental concept in probability theory and sports mathematics.
For example, in our tennis problem, the probability that a player wins a match is conditional on the probability of them winning each set.
By using the formulas provided, we can calculate the overall match-winning probabilities based on set-winning probabilities.
This helps us understand how each set outcome influences the final match result.