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.
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.
To find P((A∩B')∪(A'∩B)), we can use the principle of inclusion-exclusion.
Step 1: Calculate P(A'∩B) and P(A∩B'):
P(A'∩B) can be found using the formula P(A'∩B) = P(B) - P(A∩B), so P(A'∩B) = b - c.
P(A∩B') can be found using the formula P(A∩B') = P(A) - P(A∩B), so P(A∩B') = a - c.
Step 2: Calculate P((A∩B')∪(A'∩B)):
Using the formula for the union of two events, P((A∩B')∪(A'∩B)) = P(A∩B') + P(A'∩B) - P((A∩B')∩(A'∩B)).
P((A∩B')∩(A'∩B)) is the intersection of (A∩B') and (A'∩B), and as these events are mutually exclusive, their intersection is an empty set, so P((A∩B')∩(A'∩B)) = 0.
Therefore, P((A∩B')∪(A'∩B)) = P(A∩B') + P(A'∩B) - P((A∩B')∩(A'∩B))
= (a - c) + (b - c) - 0
= a + b - 2c.
Thus, P((A∩B')∪(A'∩B)) = a + b - 2c in terms of a, b, and c.
To find P ((A∩B′)∪(A′ ∩B)) in terms of a, b, and c, we can break it down step by step.
1. Start with (A∩B′): This represents the intersection of A and the complement of B. The complement of B (B′) is the set of all elements in the sample space that are not in B.
2. Next, (A′ ∩B): This represents the intersection of the complement of A (A′) and B.
Now, to find the probability of the union of these two events, (A∩B′)∪(A′ ∩B), we need to consider the probability of each event separately and subtract the overlapping event (A∩B).
For (A∩B′):
- P(A∩B′) = P(A) - P(A∩B)
- P(A∩B′) = a - c
For (A′ ∩B):
- P(A′ ∩B) = P(B) - P(A∩B)
- P(A′ ∩B) = b - c
Now, to find the probability of the union:
P ((A∩B′)∪(A′ ∩B)) = P(A∩B′) + P(A′ ∩B) - P(A∩B)
Substituting the values we found above:
P ((A∩B′)∪(A′ ∩B)) = (a - c) + (b - c) - c
Simplifying further:
P ((A∩B′)∪(A′ ∩B)) = a + b - 2c
Therefore, P ((A∩B′)∪(A′ ∩B)) = a + b - 2c in terms of a, b, and c.