Answer: P(B) is equal to 0.39.
Explanation:
To find P(B), we can use the complement rule and the formula for the probability of the union of two events.
1. P((A ∪ B)') = 1 - P(A ∪ B)
Since (A ∪ B)' represents the complement of (A ∪ B), we can rewrite the equation as:
1 - P(A ∪ B) = 0.39
2. P(A ∩ B) = 0 - 1
Since the given value is negative, we can rewrite it as:
P(A ∩ B) = -1
3. P(B \ A) = 2P(A \ B)
This equation tells us that the probability of B occurring without A is twice the probability of A occurring without B.
Using these equations, we can solve for P(B).
Let's start with equation 3:
P(B \ A) = 2P(A \ B)
Since P(A \ B) represents the probability of A occurring without B, and P(B \ A) represents the probability of B occurring without A, we can rewrite equation 3 as:
P(B) = 2P(A \ B)
Now, let's substitute this expression for P(B) into equation 1:
1 - P(A ∪ B) = 0.39
Substituting P(B) = 2P(A \ B):
1 - (P(A) + P(B) - P(A ∩ B)) = 0.39
Substituting P(A ∩ B) = -1 and rearranging the equation:
1 - (P(A) + 2P(A \ B) + 1) = 0.39
Simplifying the equation:
-2P(A \ B) = -0.39
Dividing both sides by -2:
P(A \ B) = 0.39/2
P(A \ B) = 0.195
Now, substituting this value back into P(B) = 2P(A \ B):
P(B) = 2 * 0.195
P(B) = 0.39