To solve this problem, we can use the concept of set theory and probability.
Step 1: Let's break down the given information using the properties of set theory and probability.
Given:
P(A ∩ B) = 0.2
P(A ∪ B) = 0.6
P(B ∪ ̄A) = 0.8
Step 2: We can use the formula for the probability of the union of two events to find P(A ∪ B).
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Given that P(A ∪ B) = 0.6 and P(A ∩ B) = 0.2, we can substitute these values in the equation and solve for P(A ∪ B):
0.6 = P(A) + P(B) - 0.2
Step 3: Rearrange the equation to solve for P(A):
P(A) = 0.6 - P(B) + 0.2
Step 4: Now, we need to find P(B ∪ ̄A). The complement of event A is denoted by ̄A, which represents all the outcomes in the sample space S that are not in A.
Using the formula for the probability of the union of two events, we have:
P(B ∪ ̄A) = P(B) + P(̄A) - P(B ∩ ̄A)
We can simplify this equation to:
P(B ∪ ̄A) = P(B) + P(̄A) - P(B ∩ ̄A) = 0.8
Step 5: Since we want to find P(A) and P(B), we need to eliminate P(̄A) and P(B ∩ ̄A). To do this, we can use the complement rule and rewrite P(B ∪ ̄A) in terms of P(A ∪ B) as follows:
P(B ∪ ̄A) = 1 - P(A ∪ B)
Substituting the given value of P(A ∪ B) = 0.6, we have:
0.8 = 1 - 0.6
Step 6: Now we can solve for P(B):
0.8 = 1 - P(A ∪ B) + P(B) = 0.4 + P(B)
P(B) = 0.8 - 0.4 = 0.4
Step 7: Finally, substitute the value of P(B) into the equation for P(A) that we obtained earlier:
P(A) = 0.6 - P(B) + 0.2 = 0.6 - 0.4 + 0.2 = 0.4
Answer:
(a) P(A) = 0.4
(b) P(B) = 0.4