a) Find P (A ∩ B′ ) when P (A ∪ B) = 0.8, P (A) = 0.5 and P (B) = 0.4.
To find P(A ∩ B'), we need to find the intersection of event A and the complement of event B.
To start with, we know that P(A ∪ B) = 0.8. This means that the probability of either event A or event B or both occurring is 0.8.
We also know that P(A) = 0.5 and P(B) = 0.4. The probability of event A occurring is 0.5, and the probability of event B occurring is 0.4.
Now we can use these probabilities to find P(A ∩ B'). Recall that the complement of event B (denoted as B') is the set of all outcomes that are not in event B.
To find P(A ∩ B'), we can use the formula:
P(A ∩ B') = P(A) - P(A ∩ B)
First, let's find P(A ∩ B). The probability of the intersection of event A and event B is given by:
P(A ∩ B) = P(A) + P(B) - P(A ∪ B)
Substituting the known values into the formula, we have:
P(A ∩ B) = 0.5 + 0.4 - 0.8
Simplifying, we get:
P(A ∩ B) = 0.9 - 0.8
P(A ∩ B) = 0.1
Now, we can use this value to find P(A ∩ B'):
P(A ∩ B') = P(A) - P(A ∩ B)
Substituting the known values into the formula, we have:
P(A ∩ B') = 0.5 - 0.1
P(A ∩ B') = 0.4
Therefore, P(A ∩ B') = 0.4.