Understanding Conditional Probability Relationships

This question explores the relationship between conditional probabilities when the occurrence of one event influences the probability of another. We are given a condition involving P(A|B) and P(A) and need to determine the correct implication for P(B|A) and P(B).

Key Concepts: Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. The formula for the conditional probability of event A given event B is:

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

Similarly, the conditional probability of event B given event A is:

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

Here, $P(A \cap B)$ represents the probability that both events A and B occur.

Analyzing the Given Condition: P(A|B) < P(A)

We are given that the probability of A occurring given B has occurred is less than the probability of A occurring normally. This means that the occurrence of event B makes event A less likely.

The given condition is:

$$ P(A|B) < P(A) $$

Using the formula for $P(A|B)$, we can substitute:

$$ \frac{P(A \cap B)}{P(B)} < P(A) $$

Assuming $P(B) > 0$ (which must be true for $P(A|B)$ to be defined), we can multiply both sides by $P(B)$:

$$ P(A \cap B) < P(A) \cdot P(B) $$

This inequality tells us that the probability of both A and B happening together is less than the product of their individual probabilities. This indicates that events A and B are not independent; specifically, they are negatively dependent in the sense that one event makes the other less likely.

Determining the Relationship for P(B|A)

Now, let's look at the expression for $P(B|A)$:

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

We want to compare $P(B|A)$ with $P(B)$. We already established that $P(A \cap B) < P(A) \cdot P(B)$.

Let's substitute the inequality for $P(A \cap B)$ into the expression for $P(B|A)$:

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

Since $P(A \cap B) < P(A) \cdot P(B)$, we can write:

$$ P(B|A) < \frac{P(A) \cdot P(B)}{P(A)} $$

Assuming $P(A) > 0$ (which must be true for $P(B|A)$ to be defined), we can cancel $P(A)$ from the numerator and the denominator:

$$ P(B|A) < P(B) $$

This result shows that if $P(A|B) < P(A)$, then it must also be true that $P(B|A) < P(B)$. This means that if the occurrence of event B makes event A less likely, then the occurrence of event A also makes event B less likely.

Checking the Options

Let's compare our finding with the given options:

• Option 1: $P(B|A) < P(B)$ - This matches our derived result.
• Option 2: $P(B|A) > P(B)$ - This is the opposite of our result.
• Option 3: $P(B|A) = P(B)$ - This would imply independence, which is not the case when $P(A|B) < P(A)$.
• Option 4: $P(B|A) > P(A)$ - This compares $P(B|A)$ to $P(A)$, which is not the direct relationship we derived from the given condition.

Therefore, the correct option is the one stating that $P(B|A) < P(B)$.

Summary of Probability Relationship

We started with the condition that $P(A|B) < P(A)$ and through algebraic manipulation using the definition of conditional probability, we proved that this implies $P(B|A) < P(B)$.

Revision Table: Probability Concepts

Additional Information: Event Dependence

Two events, A and B, are considered statistically independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, this is expressed in several equivalent ways:

• $P(A \cap B) = P(A) \cdot P(B)$
• $P(A|B) = P(A)$ (if $P(B) > 0$)
• $P(B|A) = P(B)$ (if $P(A) > 0$)

If any of these conditions do not hold, the events are considered statistically dependent. The problem illustrates a case of dependence, specifically negative dependence, where the conditional probability is less than the marginal probability.

The relationship $P(A|B) < P(A)$ implying $P(B|A) < P(B)$ shows that the nature of dependence is symmetric. If B negatively influences A, A also negatively influences B.