Problem 31 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Team \(A\) has probability \(2 / 3\) of winning whenever it plays. Suppose \(A\) plays 4 games; then the probability that \(A\) wins more than half of its games is (a) \(16 / 27\) (b) \(19 / 27\) (c) \(19 / 81\) (d) \(32 / 81\)

Step-by-Step Solution

Answer

The probability that Team A wins more than half of its games is \(\frac{16}{27}\) (option a).

Step 1: Define the Problem

We need to find the probability that Team \(A\) wins more than half of its games in 4 matches. This means winning at least 3 games out of 4.

Step 2: Define Winning Scenarios

Winning more than half means Team \(A\) wins either 3 or 4 games out of 4. Let's calculate the probabilities for both scenarios.

Step 3: Calculate Probability of Winning 3 Games

Use the binomial probability formula: \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \). Here, \( n = 4 \), \( k = 3 \), and \( p = \frac{2}{3} \). \( P(X = 3) = \binom{4}{3} \left(\frac{2}{3}\right)^3 \left(\frac{1}{3}\right)^1 \).

Step 4: Calculate Probability for Winning 4 Games

Again using the binomial probability formula for \( k = 4 \): \( P(X = 4) = \binom{4}{4} \left(\frac{2}{3}\right)^4 \left(\frac{1}{3}\right)^0 \).

Step 5: Sum the Probabilities

Sum the probabilities from Steps 3 and 4: \( P(X = 3) + P(X = 4) \) to get the probability that Team \(A\) wins more than half of its games.

Step 6: Calculate Values

\( \binom{4}{3} = 4 \), so \( P(X = 3) = 4 \times \left(\frac{8}{27}\right) \times \frac{1}{3} = \frac{32}{81} \).\( \binom{4}{4} = 1 \), so \( P(X = 4) = 1 \times \left(\frac{16}{81}\right) = \frac{16}{81} \).

Step 7: Final Calculation

The total probability is \( P(X = 3) + P(X = 4) = \frac{32}{81} + \frac{16}{81} = \frac{48}{81} = \frac{16}{27} \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with calculating the likelihood of different outcomes. It provides a quantitative description of uncertainty and helps us measure the chances that a particular event will happen. In the context of games or experiments, probability estimates how likely we are to observe a particular result.
In this exercise, we're particularly interested in the probability that Team A wins more than half of its games. The outcome of each game is uncertain, which is where probability theory comes into play. The probability of an event, such as winning a game, is expressed as a ratio or fraction ranging from 0 to 1. A probability of 0 indicates impossibility, while a probability of 1 indicates certainty.
The concept of probability is foundational in statistics and has practical applications in various fields, from sports and finance to computer science and engineering. In solving problems like this exercise, understanding probability theory allows students to predict outcomes of random events based on known probabilities.

Combinatorics

Combinatorics is a field of mathematics focused on counting, arranging, and finding patterns. In probability, it helps us determine the number of possible ways an event can occur, which is essential when dealing with complex problems.
In this particular exercise, we use combinatorics to determine how many ways Team A can win exactly 3 or 4 games out of 4. The key tool from combinatorics used here is the binomial coefficient, represented as \( \binom{n}{k} \). This coefficient tells us how many ways we can choose \( k \) successes (like winning a game) from \( n \) trials (total games played).
Here's what it looks like:

• \( \binom{4}{3} = 4 \) indicates there are 4 ways to win 3 games out of 4.
• \( \binom{4}{4} = 1 \) indicates there is exactly 1 way to win all 4 games.

Using combinatorics simplifies the process of solving probability equations, especially when dealing with multiple trials, as seen in this problem.

Binomial Distribution

The binomial distribution is a specific probability distribution that sums up the likelihood of a given number of successes in a fixed number of trials. Each trial must have two possible outcomes, often framed as "success" or "failure." For instance, in this exercise, "winning" is a success for Team A.
The binomial probability formula, \( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \), helps us calculate the probability of getting exactly \( k \) successes (wins) out of \( n \) trials (games). In our exercise:

• \( n = 4 \) and \( p = \frac{2}{3} \) (the probability of winning a single game).
• The formula calculates the probabilities of winning 3 games \( P(X=3) \) and all 4 games \( P(X=4) \).

When talking about "more than half," we combine these results: \( P(X \geq 3) = P(X=3) + P(X=4) \). The sum gives us the overall probability that Team A wins more than half its games. This use of the binomial distribution is vital as it gives a structured approach to handle problems with multiple random events."