Problem 149 - A Complete Resource Book in Mathematics for JEE Main

Question

The probability of India winning a test match against West-Indies is \(1 / 2\) assuming independence from match to match. The probability that in a match series India's second win occurs at the third test is : (A) \(\frac{1}{8}\) (B) \(\frac{1}{4}\) (C) \(\frac{1}{2}\) (D) \(\frac{1}{3}\)

Step-by-Step Solution

Answer

(A) \( \frac{1}{8} \)

Step 1: Understanding the Condition

We need to calculate the probability that India wins exactly two matches out of the first three, with the second win specifically in the third test. This implies that India must win the first test, either lose or win the second test, and win the third test.

Step 2: Determine the Sequence of Wins and Losses

The sequence of India winning exactly two matches with the second win in the third test is: **Win, Lose, Win** (WLW).

Step 3: Calculating the Probability for the Sequence

Calculate the probability for the sequence WLW: - Probability of winning: \( P(W) = \frac{1}{2} \)- Probability of losing: \( P(L) = \frac{1}{2} \)Thus, the probability for the WLW sequence is:\[ P(WLW) = P(W) \cdot P(L) \cdot P(W) = \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) \cdot \left(\frac{1}{2}\right) = \frac{1}{8} \]

Step 4: Concluding the Probability

The probability that India's second win occurs at the third test, given the sequence of WLW, is calculated to be \( \frac{1}{8} \). Therefore, the correct answer is option (A).

Key Concepts

Test Match

A Test Match is a format of cricket played between two international teams over a period of up to five days. Unlike shorter forms of cricket, test matches are known for their length and for the strategy involved, with each team having two innings to score runs. This form of cricket allows for a wide array of statistical analysis and strategic planning.
Test matches provide opportunities to observe probabilities and outcomes under various conditions. The independence of matches, as in this exercise, showcases how one result doesn't influence another directly. Each match is treated as a separate probability event in its own right.

• The length of the game adds complexity and depth to the analysis.
• More opportunities for swings in probability due to match conditions and player form.
• Each match outcome remains unaffected by those before it.

This setting lays down the foundation for exercises involving probability sequences like the one we are exploring here.

Winning Sequence

A winning sequence in a series of matches refers to the particular order of wins and losses needed to achieve a specific outcome. Understanding this is vital, as many probability problems are based on setting the conditions for this sequence.
A clear example is the sequence **WLW**, where India wins the first test, does not necessarily win the second, but must win the third for their second overall win to occur there.
In probability terms:

• The sequence of wins and losses directly impacts the probability.
• Predicting outcomes requires pinpointing the exact match for each required win.

This precise order makes the problem unique and challenges students to consider combinations of results that satisfy given conditions while focusing heavily on strategy.

Probability Calculation

Probability Calculation involves computing the likelihood of certain outcomes occurring, which in this exercise, is based on a sequence of events. Calculating probabilities in sequences like **WLW** employs multiplication of the probabilities of individual matches.

Each outcome is considered independently:

• Probability of a win, denoted as \( P(W) \), is \( \frac{1}{2} \).
• Probability of a loss, \( P(L) \), is also \( \frac{1}{2} \).
• The sequence probability: \[ P(WLW) = P(W) \cdot P(L) \cdot P(W) \]

Thus, each step is calculated with care to combine the odds correctly, leading to the final probability of \( \frac{1}{8} \). This approach allows for a deep understanding of the dependency across matches while maintaining the independence of single events.