Problem 19 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

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 5 match series India's second win occurs at the third test is (a) \(2 / 3\) (b) \(1 / 2\) (c) \(1 / 4\) (d) \(1 / 8\)

Step-by-Step Solution

Answer

The correct answer is not listed among the options.

Step 1: Understanding the Problem

We need to find the probability that India's second win occurs in the third test match of a 5-match series. The probability of winning each match is \( \frac{1}{2} \).

Step 2: Conceptualizing the Scenario

For India to have its second win in the third match, they must win exactly two of the first three matches, and specifically, the third match must be a win.

Step 3: Probability of Winning Two of Three Matches

We calculate the probability of exactly two wins in three matches using combinations. There are three possible winning combinations for two matches: (W, W, L), (W, L, W), and (L, W, W), where the third match is a win.

Step 4: Calculating Individual Probability

The probability of each sequence (W, W, L), (W, L, W), or (L, W, W) is \( \left( \frac{1}{2} \right)^3 \), as each match's outcome is independent and has probability \( \frac{1}{2} \).

Step 5: Adding Probabilities for Favorable Outcomes

Since there are three favorable sequences, we multiply the individual probability by 3: \[3 \times \left( \frac{1}{2} \right)^3 = 3 \times \frac{1}{8} = \frac{3}{8}.\]

Step 6: Ensuring No More Wins in Remaining Matches

India cannot win more than once in the remaining two matches, i.e., it can win 0 or 1 match in these. Here, only the sequence (L, L) is viable; its probability is \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \).

Step 7: Combining All Conditions

The overall probability is the product of winning two of the first three matches, with the third being a win, and not exceeding one win in the last matches: \[ \frac{3}{8} \times \frac{1}{4} = \frac{3}{32}.\]

Key Concepts

Independent Events

In probability, independent events are scenarios where the outcome of one event does not affect the outcome of another. This means that the result of one match in a series is not influenced by the results of previous or following matches. Here, each cricket match between India and West Indies is considered an independent event. This is crucial because it allows us to use the same probability for each match.

For example, if the chance of winning a single match is \( \frac{1}{2} \), then the chance remains \( \frac{1}{2} \) for each subsequent match, regardless of the outcomes of the matches already played.

• Independent events allow us to simply multiply probabilities of individual events to find the overall probability of a sequence of independent events.
• This concept is pivotal when calculating probabilities in series or sequences.

Understanding this enables us to tackle more complex probability scenarios by keeping calculations manageable and consistent.

Combinatorics in Probability

Combinatorics deals with counting combinations and arrangements, vital for calculating probabilities in sequences and series. In the exercise, we use combinatorics to determine the number of ways India can win exactly two of the first three matches in a 5-match series.

Each of these matches has specific outcomes mapped as (W, W, L), (W, L, W), and (L, W, W). Combinatorics helps list these possible outcomes, and each of these sequences aligns with the requirement that the third match is a win.

• For finding probabilities, determine all possible favorable sequences.
• The number of ways to arrange 'wins' and 'losses' helps us calculate the overall probability.

In this exercise, combinatorial counting led us to identify three ways to win two out of three matches, making it essential for completing probability calculations correctly.

Conditional Probability

Conditional probability is about finding the probability of an event given that another event has already occurred. While this concept was more implicit than explicit in the original exercise, it underlies the logic of ensuring India wins exactly two matches, with the second win specifically in the third match.

The condition is established by defining the sequence order of wins. Once India has two wins within the three initial matches with a specific third match win, we're effectively using conditional reasoning by then limiting the outcomes further and evaluating the probability of zero or one additional win.

• Conditional probability refines outcomes based on prior events’ occurrence.
• It plays a hidden, yet crucial role in layered calculations and sequence restrictions.

Understanding conditional probability helps in creating a structured approach, focusing only on the scenarios where specific conditions, such as winning the third match as the second win, have been satisfied.