Problem 122 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(A\) and \(B\) are two events. Odds against \(A\) are \(2: 1\). Odds in favour of \(A \cup B\) are \(3: 1\). If \(x \leq P(B) \leq y\), then (A) \(x=\frac{5}{12}\) (B) \(x=\frac{3}{4}\) (C) \(y=\frac{5}{12}\) (D) \(y=\frac{3}{4}\)

Step-by-Step Solution

Answer

(A) \(x=\frac{5}{12}\)

Step 1: Determine Probability of Event A

The odds against event \(A\) are given as \(2:1\). This means that the probability \(P(A)\) is given by the formula \( \frac{1}{2+1} = \frac{1}{3} \).

Step 2: Calculate Probability of A Complement

The probability that event \(A\) does not occur, denoted as \(P(A')\), is equal to \(1 - P(A)\). Therefore, \(P(A') = 1 - \frac{1}{3} = \frac{2}{3}\).

Step 3: Determine Probability of A Union B

The odds in favor of \(A \cup B\) are \(3:1\). This means the probability of \(A \cup B\) is \( \frac{3}{3+1} = \frac{3}{4}\).

Step 4: Use Inclusion-Exclusion Principle

According to the principle of inclusion-exclusion, \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Simplifying the inequality \(\frac{1}{3} + P(B) - (0) = \frac{3}{4}\), since \(A \cap B\) is not relevant in this context, results in \( P(B) = \frac{3}{4} - \frac{1}{3} = \frac{5}{12} \).

Step 5: Examine Range of P(B) with Constraints

Since \(P(B)\) must satisfy the inequality \(x \leq P(B) \leq y\), and we established that \(P(B) = \frac{5}{12}\), it follows that the range is exactly at the limit \(\frac{5}{12}\). This means \(x = \frac{5}{12}\) and \(y = \frac{5}{12}\).

Step 6: Match Final Values with Options

From the options given, \(x = \frac{5}{12}\) matches with option (A), and \(y = \frac{5}{12}\) doesn't exactly match any single option since it meets both constraints.

Key Concepts

Odds

Odds are a way to express the likelihood of an event happening compared to it not happening. When we talk about the odds against an event, like event \(A\) in our problem, we use the format \(2:1\), meaning that for every 2 times the event doesn't happen, it happens once. To find the probability from the odds, we sum the numbers in the odds and place the number of outcomes that favor the event happening over this sum. For event \(A\) with odds \(2:1\), the probability \(P(A)\) becomes \(\frac{1}{2+1} = \frac{1}{3}\).
This is because we have 1 favorable outcome out of a total of 3 possible outcomes. Remember, odds can also be presented as odds in favor. Here, the calculation is similar, but you're emphasizing how likely the event is to happen rather than to not happen.

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle helps us to find the probability of the union of two events by considering their individual probabilities and the overlap between them, which is their intersection. The principle is given by the formula:

• \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

In the context of our exercise, we want to find \(P(A \cup B)\), given the individual probabilities. The key here is that if there is no overlap or intersection (which is treated as \(0\) in this case), the inclusion-exclusion formula simplifies.
We calculated \(P(A \cup B)\) based on the given odds, resulting in \(\frac{3}{4}\). Knowing \(P(A) = \frac{1}{3}\), we plugged these into the formula to isolate \(P(B)\), leading us to \(P(B) = \frac{5}{12}\). This uses the principle to successfully determine unknown probabilities given partial information.

Set Theory

Set Theory forms the mathematical underpinning for dealing with groups of elements, exemplified here with events \(A\) and \(B\). When you see symbols such as \(\cup\) and \(\cap\), think of them as operations on these groups or sets. The union, \(A \cup B\), represents the set of outcomes that are in \(A\), in \(B\), or in both.
Meanwhile, \(A \cap B\) stands for the intersection, or the set of outcomes present in both \(A\) and \(B\). In probability, these operations translate into calculations of likelihoods. For example, when we used the union in our exercise, we applied it to calculate \(P(A \cup B)\), leveraging the given odds to reveal relationships between probabilities.
Mastering set theory basics is crucial in understanding more intricate probability scenarios, as it equips you to handle combinations and intersections with ease.