Problem 15 - A Complete Resource Book in Mathematics for JEE Main 2017

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 \(\mathrm{S}_{1}\) 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 that player \(S_1\) is among the eight winners is \(\frac{1}{2}\).

Step 1: Understanding the Problem

We need to calculate the probability that player \(S_1\) is one of the eight winners when sixteen players are paired at random, and the winner from each pair is decided by equal probability. Each player has an equal chance of winning if selected in the same pair.

Step 2: Identify the Pair Scenario

When sixteen players are paired, there are a total of 8 pairs. Each player is paired with another randomly. We're interested in the pair containing \(S_1\).

Step 3: Calculate Possible Outcomes for Pair

In the pair containing \(S_1\), there are only two players, \(S_1\) and the other player. In this situation, since both players have equal skill levels, \(S_1\) has a \(\frac{1}{2}\) chance of winning.

Step 4: Conclude with the Required Probability

Since \(S_1\) has a \(\frac{1}{2}\) probability of winning in the pair it's in, the probability that \(S_1\) is among the eight winners is also \(\frac{1}{2}\). Therefore, the correct choice of the probability that \(S_1\) is among the eight winners is (A) \(\frac{1}{2}\).

Key Concepts

Equal Strength Players

In the world of probability and game theory, the assumption that players have equal strength simplifies many calculations. When we say players are of equal strength, we mean that each player has the same skill level and the same likelihood of winning or losing any given match. This creates fairness and removes any bias in determining outcomes.
For example, if two players with equal strength face off, each has a 50% chance of winning. In our exercise, 16 players participate in a tournament, but their equal abilities level the playing field, making the analysis straightforward. This assumption of equal strength is crucial, as it justifies why each outcome in pairings is equally probable, precisely what we need for the calculation of the probability.

Random Pairing

Random pairing is a method used to ensure fairness in tournaments or competitions. When participants are paired randomly, it means that no pattern or preference determines who competes against whom. This randomness is essential because it helps minimize biases or pre-conceived advantages, leading to a fairer contest.
In our exercise, the 16 players are paired randomly, meaning that any player can pair with any other player with equal likelihood. This randomness implies that player \(S_1\) can be in a pair with any of the other 15 players, making the game unpredictable but fair. Random pairing is important in game theory and probability because it maintains the integrity of chance and competition.

Game Theory

Game theory is the study of strategic interactions where the outcome for each participant depends on the actions of all involved. It offers insights into competitive situations and helps analyze potential strategies for players. In contexts like our exercise, where players are randomly paired, game theory examines how these setups influence probabilities and outcomes.
Given the players have equal strength and are paired randomly, game theory suggests that each has an equal probability of winning their match as their opponent. Through game theory, we understand how these dynamics and rules affect the likelihood of any player, such as \(S_1\), emerging as a winner. This forms the basis for predicting outcomes based on strategic gameplay.

Winner Selection

Winner selection is the process of determining the victor in a competitive setup. In our exercise, after random pairing, a winner is decided from each pair. With all players having equal strength, this decision relies solely on chance, giving each participant a 50% probability of winning.
When the player \(S_1\) is paired against any other player, since all are equally skilled, the probability of \(S_1\) winning is \(\frac{1}{2}\). This 50-50 chance is fundamental in probabilistic models and defines the fairness of the tournament setup. The approach ensures that the winner selection is unbiased and purely based on luck, reflecting the foundational principles of probability and fairness.