Q.If A and B are two candidates seeking admission in an engineering College. The probability that A is selected is $.5$ and the probability that both A and B are selected is at most $.3$. Is it possible that the probability of B getting selected is $0.7$?

The key idea here is the Probability Axioms, specifically the property that the probability of the intersection of two events cannot exceed the probability of either individual event.

1. We are given $P(A) = 0.5$ and $P(A \cap B) \le 0.3$. We need to check if $P(B) = 0.7$ is possible.
2. A fundamental property of probability states that for any two events A and B, $P(A \cap B) \le P(A)$ and $P(A \cap B) \le P(B)$.
3. Using the given $P(A) = 0.5$, we must have $P(A \cap B) \le 0.5$.
4. If $P(B) = 0.7$ were true, we would also require $P(A \cap B) \le 0.7$. …

In probability, we operate under a set of fundamental rules, known as axioms, that ensure consistency and logical coherence. One crucial axiom states that the probability of any event must be between $0$ and $1$ (inclusive). Another important derived rule is the Addition Rule for two events, which relates the probabilities of individual events to the probability of their union and intersection.

The question asks if a specific probability for event B is possible given other constraints. To determine possibility, we need to check if assuming $P(B) = 0.7$ leads to any contradiction with the fundamental axioms or the given information. If we can find a scenario where all conditions are met, then it is possible.

Let's define the events:

• $A$: Candidate A is selected.
• $B$: Candidate B is selected.

We are given the following probabilities:

• $P(A) = 0.5$
• $P(A \cap B) \le 0.3$ (The probability that both A and B are selected is at most $0.3$)

We need to determine if it is possible that $P(B) = 0.7$.

1. Recall the Addition Rule for Probabilities: For any two events $A$ and $B$, the probability of their union (either A or B or both are selected) is given by:

2. Apply the Axiom of Total Probability: A fundamental axiom of probability states that the probability of any event cannot exceed $1$. Therefore, the probability of the union of A and B must also be less than or equal to $1$:

Combining this with the Addition Rule, we get:

3. Substitute the given and hypothetical values: Let's assume $P(B) = 0.7$ and substitute the given $P(A) = 0.5$ into the inequality:

4. Determine the implied lower bound for $P(A \cap B)$: Rearranging the inequality to isolate $P(A \cap B)$:

This means that if $P(B) = 0.7$, then the probability of both A and B being selected must be at least $0.2$. …