EXERCISE: THE MULTIPLICATION RULE (4 points possible)

Are the following statements true or false? (Assume that all conditioning events have positive probability.)

P(A∩B∩Cc)=P(A∩B)P(Cc∣A∩B)

Looking at a Venn diagram I suspect true

To determine whether the statement is true or false, we can use the multiplication rule of probability.

The multiplication rule states that for any two events A and B, the probability of both events occurring, denoted as P(A∩B), can be found by multiplying the probability of A by the conditional probability of B given A. Mathematically, this can be written as:

P(A∩B) = P(A) * P(B|A)

In the given statement, we have:

P(A∩B∩Cc) = P(A∩B) * P(Cc|A∩B)

Here, Cc represents the complement of event C, which refers to the event that C does not occur.

To determine whether the statement is true or false, we need to compare the left-hand side (LHS) and right-hand side (RHS) of the equation.

If the LHS of the equation is equal to the RHS, then the statement is true. Otherwise, it is false.

So, to determine the truth value of the statement, we need to calculate both the LHS and RHS of the equation using the given information and probability rules.