For two dependent events A and B, it is given that P(A) = 0.2 and P(B) = 0.5. If A ⊆ B, then the values of conditional probabilities P(A|B) and P(B|A) are respectively
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.
| Conditional Probability | Formula | Calculation | Value |
|---|---|---|---|
| P(A|B) | \(\frac{P(A \cap B)}{P(B)}\) | \(\frac{P(A)}{P(B)} = \frac{0.2}{0.5}\) | \(\frac{2}{5}\) |
| P(B|A) | \(\frac{P(A \cap B)}{P(A)}\) | \(\frac{P(A)}{P(A)}\) | 1 |
Revision Table: Key Concepts in Probability
| Concept | Description | Formula/Notation |
|---|---|---|
| Probability of an Event | The likelihood of an event occurring. | P(E) |
| Dependent Events | Events where the outcome of one affects the outcome of the other. | P(A ∩ B) \(\neq\) P(A)P(B) |
| Conditional Probability | The probability of an event occurring given that another event has already occurred. | P(A|B) or P(B|A) |
| Intersection of Events | The event where both A and B occur. | A ∩ B (or A AND B), P(A ∩ B) |
| Subset of Events | If A ⊆ B, every outcome in A is also in B. | If A ⊆ B, then A ∩ B = A |
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.
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