Question

If the occurrence of \(B\) makes \(A\) more likely, does the occurrence of \(A\) make \(B\) more likely?

Short answer

In conclusion, just because the occurrence of B makes A more likely, it does not imply that the occurrence of A will make B more likely. The relationship between the occurrence of A making B more likely will depend on the values of probabilities of events A and B, and the probability of both events happening together, which can vary depending on the specific scenario.

Step-by-step solution

Understand conditional probability

Conditional probability is denoted by P(A|B), which means the probability of event A occurring given that event B has occurred. It is calculated using the formula: \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) Here, P(A ∩ B) is the probability of both A and B happening, and P(B) is the probability of event B happening.

Analyse the given condition

We are given that the occurrence of B makes A more likely. Mathematically, this can be represented as: \(P(A|B) > P(A)\) Which means the probability of A happening, given that B has happened, is greater than the probability of A happening without any information about B.

Determine if the occurrence of A makes B more likely

Now, we need to find if the occurrence of A makes B more likely. In other words, we need to determine whether: \(P(B|A) > P(B)\) Using the formula of conditional probability, we can rewrite this expression as: \(\frac{P(A \cap B)}{P(A)} > P(B)\) Now, multiplying both sides by P(A): \(P(A \cap B) > P(A)P(B)\) Comparing this inequality to the initial condition, we cannot conclude a definitive relationship between the occurrence of A making B more likely or not. It is because the inequality depends on the values of the probabilities, which can vary.

Conclusion

In conclusion, just because the occurrence of B makes A more likely, it does not imply that the occurrence of A will make B more likely. The relationship between the occurrence of A making B more likely will depend on the values of probabilities of events A and B, and the probability of both events happening together, which can vary depending on the specific scenario.