Let A and B be two events in a sample space S such that P(A) = 0.5, P(B) = 0.6, and P(A intersection B) = 0.15. Find the probabilities below. Hint: (A intersection Bc) union (A intersection B) = A.
(a) P(A|Bc) ________
(b) P(B|Ac) ________
To find the probabilities, we'll use the formula for conditional probability: P(A|B) = P(A intersection B) / P(B).
(a) To find P(A|Bc) (the probability of A given not B), we need to find P(A intersection Bc) and P(Bc).
First, we know that (A intersection Bc) union (A intersection B) = A. Rearranging this equation gives us P(A intersection Bc) = P(A) - P(A intersection B) = 0.5 - 0.15 = 0.35.
To find P(Bc) (the probability of not B), we can use the fact that P(Bc) = 1 - P(B) = 1 - 0.6 = 0.4.
Now we can calculate P(A|Bc) using the formula for conditional probability:
P(A|Bc) = P(A intersection Bc) / P(Bc)
= 0.35 / 0.4
= 0.875
Therefore, P(A|Bc) = 0.875.
(b) To find P(B|Ac) (the probability of B given not A), we'll calculate P(B intersection Ac) and P(Ac).
We can use the fact that P(Ac) (the probability of not A) = 1 - P(A) = 1 - 0.5 = 0.5.
To find P(B intersection Ac), we can use the formula (A intersection B) union (Ac intersection B) = B. Rearranging this equation gives us P(B intersection Ac) = P(B) - P(A intersection B) = 0.6 - 0.15 = 0.45.
Now we can calculate P(B|Ac) using the formula for conditional probability:
P(B|Ac) = P(B intersection Ac) / P(Ac)
= 0.45 / 0.5
= 0.9
Therefore, P(B|Ac) = 0.9.
To find the probabilities, we'll use the formulas:
(a) P(A|Bc) = P(A intersection Bc) / P(Bc)
(b) P(B|Ac) = P(B intersection Ac) / P(Ac)
First, let's find P(Bc) and P(Ac).
Given: P(A) = 0.5, P(B) = 0.6, and P(A intersection B) = 0.15.
We know that:
P(Ac) = 1 - P(A)
P(Bc) = 1 - P(B)
(a) P(A|Bc):
To find P(A|Bc), we need to find P(A intersection Bc) and P(Bc).
P(Bc) = 1 - P(B) = 1 - 0.6 = 0.4
Now, P(A intersection Bc) can be calculated using the formula:
P(A intersection Bc) = P(A) - P(A intersection B)
P(A intersection Bc) = 0.5 - 0.15 = 0.35
So, P(A|Bc) = P(A intersection Bc) / P(Bc) = 0.35 / 0.4 = 7/8 or 0.875
Therefore, P(A|Bc) = 7/8 or 0.875.
(b) P(B|Ac):
To find P(B|Ac), we need to find P(B intersection Ac) and P(Ac).
P(Ac) = 1 - P(A) = 1 - 0.5 = 0.5
Now, P(B intersection Ac) can be calculated using the formula:
P(B intersection Ac) = P(B) - P(A intersection B)
P(B intersection Ac) = 0.6 - 0.15 = 0.45
So, P(B|Ac) = P(B intersection Ac) / P(Ac) = 0.45 / 0.5 = 9/10 or 0.9
Therefore, P(B|Ac) = 9/10 or 0.9.
Now that we have calculated the probabilities, the answers are:
(a) P(A|Bc) = 7/8 or 0.875.
(b) P(B|Ac) = 9/10 or 0.9.
a,.4
b. .4