Suppose for this problem that Pr[E]=13/24 and Pr[F]=5/8

and Pr[(E U F)'] = 0. Finding Pr[E/F] and Pr[F/E].

For this problem I added 13/24 + 5/8 and got 1.17, therefore, I assumed that .17 is the intersection between Pr(E) and Pr(F). I know that to solve these problems I have to know the intersection and I know the basic formula is: Pr(E/F) = Pr(E intersected with F)/Pr(F) and vice versa.

Somewhere I am making a mistake. Can someone please help me solve this?

the answer is 8

To solve this problem, let's first clarify some concepts.

In probability theory, the intersection of two events E and F is denoted as E ∩ F, and it represents the event where both E and F occur simultaneously.

The formula you mentioned, Pr(E/F) = Pr(E ∩ F) / Pr(F), gives the probability of event E given that event F has occurred. Similarly, Pr(F/E) = Pr(F ∩ E) / Pr(E) gives the probability of event F given that event E has occurred.

Now, let's find Pr(E ∩ F). We know that Pr[(E ∪ F)'] = 0, where (E ∪ F)' represents the complement of the union of events E and F, i.e., the event that neither E nor F occurs.

Using the complement rule, we can rewrite Pr[(E ∪ F)'] as 1 - Pr(E ∪ F). Therefore, 1 - Pr(E ∪ F) = 0. From this, we can deduce that Pr(E ∪ F) = 1.

Now, to find Pr(E ∩ F), we can use the formula Pr(E ∪ F) = Pr(E) + Pr(F) - Pr(E ∩ F), which represents the inclusion-exclusion principle.

Substituting the given probabilities, we have: 1 = 13/24 + 5/8 - Pr(E ∩ F). Rearranging the terms, we get Pr(E ∩ F) = 9/24 = 3/8.

Now that we have Pr(E ∩ F), we can find Pr(E/F) and Pr(F/E) using the formulas mentioned earlier.

Pr(E/F) = Pr(E ∩ F) / Pr(F) = (3/8) / (5/8) = 3/5.

Pr(F/E) = Pr(F ∩ E) / Pr(E) = (3/8) / (13/24) = 9/26.

Therefore, Pr(E/F) = 3/5 and Pr(F/E) = 9/26.