Understanding the Core Condition

The question provides a specific relationship between the probabilities of two events, A and B. The given condition is:

$P(B) = P(A \text{ and } B)$

In probability notation, '$P(A \text{ and } B)$' is the same as the probability of the intersection of events A and B, denoted as $P(A \cap B)$. Therefore, the condition can be stated as:

$P(B) = P(A \cap B)$

This equality tells us that the probability of event B happening is equal to the probability of both A and B happening together. This implies that the set of outcomes for event B is entirely contained within the set of outcomes for event A. In other words, if event B occurs, event A must also occur. This relationship is represented by $B \subseteq A$.

Conditional Probability Definition

To answer the question, we need to understand the definition of conditional probability. Specifically, the probability of event A occurring given that event B has already occurred is denoted as $P(A|B)$. The formula is:

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

This formula is applicable when the probability of the condition, $P(B)$, is greater than zero ($P(B) > 0$).

Applying the Condition to Find P(A|B)

Now, we substitute the given condition, $P(A \cap B) = P(B)$, into the formula for $P(A|B)$:

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

Provided that $P(B) \ne 0$, the expression simplifies to:

$P(A|B) = 1$

This means that if event B occurs, event A is certain to occur. This aligns perfectly with our understanding that the condition $P(B) = P(A \cap B)$ implies $B \subseteq A$.

Analysis of Other Options

Let's consider why the other options are less likely or not always true:

• $P(B|A) = 1$: The formula for $P(B|A)$ is $\frac{P(A \cap B)}{P(A)}$. Substituting the condition gives $\frac{P(B)}{P(A)}$. This equals 1 only if $P(B) = P(A)$. While $B \subseteq A$ implies $P(B) \le P(A)$, it doesn't guarantee $P(B) = P(A)$.
• $P(B|A) = 0$: This would only be true if $P(A \cap B) = 0$. Since $P(B) = P(A \cap B)$, this means $P(B) = 0$. If $P(B)=0$, $P(A|B)$ is undefined.
• $P(A|B) = 0$: This requires $P(A \cap B) = 0$. Given $P(B) = P(A \cap B)$, this means $P(B) = 0$. If $P(B) = 0$, the conditional probability $P(A|B)$ is undefined. The case where $P(A|B)=1$ holds when $P(B) > 0$.

Conclusion

The condition $P(B) = P(A \text{ and } B)$ directly leads to the conclusion that $P(A|B) = 1$, assuming $P(B) > 0$. This signifies that event A occurs with certainty whenever event B occurs.