Question
Suppose that P (A) = a, P (B) = b and P (A∩B) = c. Find P ((A∩B′)∪(A′ ∩B))
in terms of a, b and c.
in terms of a, b and c.
Answers
GPT 3.5
To find P((A∩B')∪(A'∩B)), we can use the inclusion-exclusion principle.
First, let's find P(A∩B'). We know that B' (complement of B) consists of all the outcomes that are not in B. So, A∩B' will consist of all the outcomes that are in A but not in B.
P(A∩B') = P(A) - P(A∩B) = a - c
Next, let's find P(A'∩B). A' (complement of A) consists of all the outcomes that are not in A. So, A'∩B will consist of all the outcomes that are in B but not in A.
P(A'∩B) = P(B) - P(A∩B) = b - c
Finally, to find P((A∩B')∪(A'∩B)), we sum up P(A∩B') and P(A'∩B) and subtract the intersection of the two sets (A∩B):
P((A∩B')∪(A'∩B)) = P(A∩B') + P(A'∩B) - P(A∩B)
= (a - c) + (b - c) - c
= a + b - 2c
Therefore, P((A∩B')∪(A'∩B)) in terms of a, b, and c is a + b - 2c.
First, let's find P(A∩B'). We know that B' (complement of B) consists of all the outcomes that are not in B. So, A∩B' will consist of all the outcomes that are in A but not in B.
P(A∩B') = P(A) - P(A∩B) = a - c
Next, let's find P(A'∩B). A' (complement of A) consists of all the outcomes that are not in A. So, A'∩B will consist of all the outcomes that are in B but not in A.
P(A'∩B) = P(B) - P(A∩B) = b - c
Finally, to find P((A∩B')∪(A'∩B)), we sum up P(A∩B') and P(A'∩B) and subtract the intersection of the two sets (A∩B):
P((A∩B')∪(A'∩B)) = P(A∩B') + P(A'∩B) - P(A∩B)
= (a - c) + (b - c) - c
= a + b - 2c
Therefore, P((A∩B')∪(A'∩B)) in terms of a, b, and c is a + b - 2c.