Problem 149 - A Complete Resource Book in Mathematics for JEE Main 2017

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 : [2002] (A) \(\frac{1}{8}\) (B) \(\frac{1}{4}\) (C) \(\frac{1}{2}\) (D) \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{4} \).

Step 1: Understand the Scenario

India has to win exactly two matches, and we want their second win to take place specifically in the third match.

Step 2: Determine the Winning Pattern

For India's second win to occur in the third match, they must have exactly one win in the first two matches. The sequence of wins and losses before the third match can be either: WL or LW.

Step 3: Calculate the Probability for First Two Matches

For either of the mentioned sequences (WL or LW) to happen in the first two matches, calculate the probability as:\(P_1 = (1/2) \cdot (1/2) = 1/4\).

Step 4: Probability of Winning the Third Match

India must win the third match. The probability of winning any single match is:\(P_2 = 1/2\).

Step 5: Combine Probabilities

Since either of the sequences (WL or LW) is acceptable before winning the third match, calculate the total probability by:Total Probability = Probability of WL or LW + Probability of Winning Third Match.\(P = 2 \times (1/4) \times (1/2) = 2 \times 1/8 = 1/4\).

Key Concepts

Independence in Probability

In probability, independence means that the outcome of one event does not affect the outcome of another event. This concept is crucial when analyzing sequences of events, such as matches in a series. When we say that the probability of India winning a test match is \(\frac{1}{2}\) and assume that it is independent of other matches, each test result is standalone. This independence implies that whether India wins or loses one match does not influence the chances of winning the next match.
It's like flipping a fair coin where each toss gives a \(\frac{1}{2}\) probability of landing heads, irrespective of past outcomes.

• The probability of wins and losses for each new match remains constant.
• Past results do not bias or change future probabilities.

Understanding this allows us to calculate probabilities across multiple events without needing to account for conditional dependencies.

Winning Patterns

Winning patterns refer to the particular sequence of outcomes leading to a desired event. For instance, if India’s second win is in the third match, they must have exactly one win in the first two matches. The outcomes for these first two matches must be either Win-Loss (WL) or Loss-Win (LW).
This sequence requirement shapes the entire probability calculation.
Consider:

• Winning the first but losing the second match (WL pattern)
• Or losing the first and winning the second match (LW pattern)

Both patterns satisfy the condition of one win in the first two matches. Recognizing these patterns is essential because it identifies all possible paths to the desired outcome, which then aids in calculating the total probability accurately.

Calculation of Probability

The calculation of probability with distinct events often requires breaking down sequences into manageable parts. Here, we considered the scenarios where India's second win is in the third match, needing analysis of the first two matches and the third one separately.
For the first two matches, identify all valid outcomes (WL or LW). The probability for either sequence occurring is \(P_1 = \left(\frac{1}{2}\right) \times \left(\frac{1}{2}\right) = \frac{1}{4}\).

Next, account for the third match where India must win:

• Probability of third match win = \(P_2 = \frac{1}{2}\).

Finally, calculate the combined probability for the sequence leading to a second win in the third match:
Combine the probabilities of WL or LW followed by a win in the third match:
\[P = 2 \times \left(\frac{1}{4}\right) \times \left(\frac{1}{2}\right) = 2 \times \frac{1}{8} = \frac{1}{4}\]
This calculation shows how understanding each step in the sequence is fundamental to finding the probability of complex events.