Problem 16 - A Complete Resource Book in Mathematics for JEE Main

Question

Sixteen players \(S_{1}, S_{2}, \ldots S_{16}\) play in a tournament. They are divided into eight pairs at random. From each pair a winner is decided on the basis of a game played between the two players of the pair. Assuming that all the palyers are to equal strength. The probability that the players \(S_{1}\) is among the eight winners is (A) \(\frac{1}{2}\) (B) \(\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \( \frac{1}{2} \) (option A).

Step 1: Understanding the Problem

The problem involves a tournament with 16 players where each player is paired with another, and the winner is determined based on a game between the paired players. Our goal is to find the probability that player \( S_1 \) wins their match and becomes one of the eight winners.

Step 2: Analyzing Player Pairing

Each of the 16 players is paired randomly, resulting in 8 pairs. Player \( S_1 \) will face one opponent out of the remaining 15 players since all players are paired randomly.

Step 3: Calculating Winning Probability

Since all players are of equal strength, player \( S_1 \) has an equal chance of winning or losing the game against their opponent. The probability that \( S_1 \) wins their game is therefore \( \frac{1}{2} \).

Key Concepts

Understanding Random Pairing in Tournaments

In the context of our tournament, random pairing means that each player is matched with another player without any predetermined order or ranking. This type of pairing is often used to maintain fairness and unpredictability in competitions.

• Each of the 16 players in the tournament can be paired with any other player.
• This randomness ensures that no player has an inherent advantage or disadvantage based solely on their pairing.
• It's especially effective in initial rounds where players' skill levels are unknown.

To determine the probability related to random pairings, it's crucial to consider each player has an equal chance of playing against any other. In our exercise, player \( S_1 \) is randomly paired with one of the 15 remaining competitors, reflecting classic random pairing rules.

The Equal Strength Assumption

The concept of equal strength is central to tournament probability problems. It assumes that all players have the same level of skill and capability. This simplifies the analysis of outcomes by focusing on chance rather than skill.

• In the equal strength scenario, each player has a 50% chance of winning their match.
• This is expressed as a probability of \( \frac{1}{2} \) for each match outcome per player.
• Since all players have equal ability, luck becomes the sole determinant of the game's result.

In this problem, the equal strength assumption is crucial, as it ensures the probability calculation focuses solely on the game under fair conditions, eliminating any bias in chances of winning.

Tournament Analysis and Outcome Probabilities

Tournament analysis involves the examination of match outcomes based on initial conditions like random pairing and equal player strength. By understanding these factors, we can predict and calculate probabilities of potential outcomes.

• With 16 players divided into 8 pairs, each winner moves to the next stage.
• Given equal strength, the probability for each player to win their round is \( \frac{1}{2} \).
• Specifically, the probability that player \( S_1 \) becomes one of the eight winners is therefore \( \frac{1}{2} \).

The approach to solving this problem involves recognizing the importance of random pairing and equal strength, then applying simple probability calculations to understand how likely it is for any player, like \( S_1 \), to advance through the tournament.