Can someone please help me out here. I can't figure out how to do them. I need Pr(A) to do them with the formulas that I know. Please help!

Given that Pr(B) = 1/5, Pr(A|B) = 1/3 and Pr(A|B') = 1/3.

Determine:
a) Pr(A and B')
b) Pr(A)
c) Pr(B'|A)

To find the answers to the given questions, we can use basic probability formulas and rules. Let's break down each question step by step:

a) Pr(A and B'):
The probability of the intersection of two events (A and B') is given by the formula: Pr(A and B') = Pr(A|B') * Pr(B').

From the information provided, we know that Pr(A|B') = 1/3 and Pr(B') = 1 - Pr(B) = 1 - 1/5 = 4/5.

Plugging in these values into the formula, we get: Pr(A and B') = (1/3) * (4/5) = 4/15.

Therefore, Pr(A and B') is equal to 4/15.

b) Pr(A):
To determine Pr(A), we can use the law of total probability, which states that the overall probability of an event can be found by considering the probabilities of all the different ways that event can occur.

In this case, we can calculate Pr(A) by using the formula: Pr(A) = Pr(A|B) * Pr(B) + Pr(A|B') * Pr(B').

From the given information, we have Pr(A|B) = 1/3, Pr(B) = 1/5, Pr(A|B') = 1/3, and Pr(B') = 4/5.

Plugging in these values into the formula, we get: Pr(A) = (1/3) * (1/5) + (1/3) * (4/5) = 7/15.

Therefore, Pr(A) is equal to 7/15.

c) Pr(B'|A):
To find Pr(B'|A), we can use Bayes' theorem, which allows us to compute the probability of an event happening, given that another event has already occurred.

The formula for Bayes' theorem is: Pr(B'|A) = (Pr(A|B') * Pr(B')) / Pr(A).

From the given information, we know that Pr(A|B') = 1/3, Pr(B') = 4/5, and Pr(A) = 7/15.

Plugging in these values into the formula, we get: Pr(B'|A) = (1/3) * (4/5) / (7/15) = 4/7.

Therefore, Pr(B'|A) is equal to 4/7.

By following the steps and formulas outlined above, you can find the values for a), b), and c) based on the given probabilities.