Question

A tennis tournament is arranged on a straight knockout basis for \(2^{n}\) players and for each round, except the final, opponents for those still in the competition are drawn at random. The quality of the field is so even that in any match it is equally likely that either player will win. Two of the players have surnames that begin with ' \(Q^{\prime}\). Find the probabilities that they play each other (a) in the final, (b) at some stage in the tournament.

Short answer

a) \(\left(\frac{1}{2}\right)^{2n-2}\); b) \(1 - \left(\frac{1}{2}\right)^{(n-1)}\)

Step-by-step solution

Identify the Total Number of Players

Determine the total number of players which is given as \(2^n\).

Total Number of Rounds

In a knockout tournament, the number of rounds needed for there to be a winner is \(n\) because each round halves the number of competitors. i.e., the final round is the \(n^{th}\) round.

Probability of Reaching the Final

Calculate the probability of each specific player reaching the final. Since the tournament is fair and each match has a 50% chance of being won by either player, the probability of one player winning \(n-1\) matches is \(\left(\frac{1}{2}\right)^{(n-1)}\).

Probability for Two Players with Surnames Beginning with Q Reaching Final

To find the probability of both players reaching the final, multiply the probability of one reaching the final by the probability of the other reaching the final: \(\left(\frac{1}{2}\right)^{(n-1)} \times \left(\frac{1}{2}\right)^{(n-1)} = \left(\frac{1}{2}\right)^{2(n-1)}\).

Simplify the Expression for Part (a)

Simplify the expression derived: \(\left(\frac{1}{2}\right)^{2(n-1)} = \left(\frac{1}{2}\right)^{2n-2}\).

Determine Probabilities for Part (b)

Two players will eventually meet provided they both keep winning. Let’s consider each possible round where the pair hasn't already played. Calculate the probability for each round, then accumulate these probabilities. Since each pair meeting round is independent and mutually exclusive, this forms a partial geometric series with an initial term of \(\frac{1}{2}\) and common ratio also \(\frac{1}{2}\).

Calculate Sum of Probabilities for (b)

The sum of such a series is: \[ \sum_{r=1}^{n-1} \left(\frac{1}{2}\right)^r = 1 - \left(\frac{1}{2}\right)^{(n-1)} \].

Key concepts

knockout tournament

A knockout tournament is a type of competition in which players are paired to compete against each other in successive rounds. The loser of each match is eliminated from the tournament, while the winner progresses to the next round until only one player remains. This format is popular in sports such as tennis and boxing. In our specific exercise, we deal with a tennis tournament with exactly \(2^n\) players, which ensures the number of players dwindles to a single champion after \(n\) rounds.

Key points about knockout tournaments:

• Each round halves the number of remaining competitors.
• Matches in each round are random, except for purposeful seeding initially.
• The final match determines the tournament winner.

This simple but dramatic structure assures high stakes, as a single loss results in elimination.

probability theory

Probability theory is the branch of mathematics that deals with the analysis of random phenomena. In the context of our tournament, this involves calculating the likelihood of certain outcomes, like whether two players will meet during any stage. Here are some key concepts:

• Random Event: Any situation where the outcome cannot be predicted with certainty (e.g., the result of a tennis match).
• Probability: A measure of the likelihood that a given event will occur. Given as a number between 0 and 1.

In our example, each player has an equal 50% chance of winning any given match. To find the probability that two players meet in the final match, we use the independence and multiplication rule of probabilities: \[ \text{If } P(A) \text{ and } P(B) \text{ are independent events, then } P(A \text{ and } B) = P(A) \times P(B). \] This principle allows us to calculate the likelihood of both players advancing to the final round.

combinatorics

Combinatorics is a mathematical field focusing on counting, combination, and permutation of sets of elements. It plays a crucial role in determining the number of ways certain outcomes can occur within our tournament:

• Permutations: How many ways certain players can be matched up.
• Combinations: The number of ways two players can be chosen to play each other at some stage.

Knowing the exact number of permutations and combinations helps us accurately determine probabilities. For instance, the number of ways two players can meet by the final round diffuses to them each winning precisely \(n-1\) matches without encountering each other.

geometric series

A geometric series is a series of terms that each follow from the previous one by multiplying by a constant factor. In assessing the probability of two players meeting at any stage of the tournament, we use the concept of geometric series.

The probability formula derived from accumulating round-by-round meeting chances forms a geometric series with:

• First term: \( \frac{1}{2} \),
• Common ratio: \( \frac{1}{2} \).

For example, the series for our problem calculates the sum to determine the success probability across the different rounds:
\[ \text{Sum} = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1 - \frac{1}{2^{n-1}} \] This evaluates the likelihood that the two players will meet at any point in the tournament before the final.