Solving Conditional Probability for Dependent Events

The question asks us to find the conditional probabilities P(A|B) and P(B|A) for two dependent events A and B, given their individual probabilities and the relationship that event A is a subset of event B (A ⊆ B).

We are given:

• P(A) = 0.2
• P(B) = 0.5
• A ⊆ B

The condition A ⊆ B is crucial. It means that if event A occurs, event B must also occur. In terms of set theory applied to events, the intersection of A and B, denoted as A ∩ B, is simply the set A itself. Therefore, the probability of the intersection P(A ∩ B) is equal to the probability of A, P(A).

So, P(A ∩ B) = P(A) = 0.2.

Calculating P(A|B)

The formula for the conditional probability of A given B is:

\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)

Substitute the values we know: P(A ∩ B) = P(A) = 0.2 and P(B) = 0.5.

\(P(A|B) = \frac{0.2}{0.5}\)

To simplify the fraction:

\(P(A|B) = \frac{2/10}{5/10} = \frac{2}{5}\)

Calculating P(B|A)

The formula for the conditional probability of B given A is:

\(P(B|A) = \frac{P(A \cap B)}{P(A)}\)

Substitute the values we know: P(A ∩ B) = P(A) = 0.2 and P(A) = 0.2.

\(P(B|A) = \frac{0.2}{0.2}\)

Since 0.2 is not zero, we can perform the division:

\(P(B|A) = 1\)

Understanding P(B|A) = 1 when A ⊆ B

The result P(B|A) = 1 makes intuitive sense given the condition A ⊆ B. P(B|A) represents the probability that event B occurs *given that* event A has already occurred. Since A is a subset of B, every time A occurs, B must necessarily occur. Therefore, the probability of B occurring given A has occurred is certain, i.e., 1.

So, the values of P(A|B) and P(B|A) are respectively \(\frac{2}{5}\) and 1.

Revision Table: Key Concepts in Probability

Additional Information: Subset Relationship in Probability

When event A is a subset of event B (A ⊆ B), this has specific implications for probabilities:

• Any outcome that satisfies event A also satisfies event B.
• The occurrence of A guarantees the occurrence of B.
• The probability of A happening together with B is the same as the probability of A happening alone: P(A ∩ B) = P(A).
• The probability of B happening together with A is also the same as the probability of A happening alone: P(B ∩ A) = P(A).
• Since A ⊆ B, it also means P(A) \(\le\) P(B). In this problem, P(A) = 0.2 and P(B) = 0.5, which satisfies this condition.
• If A ⊆ B, then P(B|A) = 1 (assuming P(A) > 0), because if A occurs, B must occur.
• If A ⊆ B, then P(A|B) = P(A) / P(B) (assuming P(B) > 0), as calculated in the solution.

Understanding the subset relationship helps simplify problems involving conditional probabilities when one event is contained within another.