Prove that:
P(A n B^c) = P(A U B) - P(B)
and
P(A n B^c) = P(A) - P(A n B)
To prove the equality:
P(A ∩ B^c) = P(A ∪ B) - P(B)
we will use the properties of set operations and the definition of probability.
First, let's break down the left-hand side:
P(A ∩ B^c) represents the probability of the event A occurring and B not occurring.
Now, let's analyze the right-hand side:
P(A ∪ B) represents the probability of either event A or event B (or both) occurring.
P(B) represents the probability of event B occurring.
To prove the equality, we need to show that the probability of A occurring when B does not occur is equal to the probability of (A or B) minus the probability of B.
Proof:
1. Start with the assumption that A and B are two events.
2. Suppose we define a new event, C, which represents the intersection of A and B.
3. By using the properties of set operations, we can rewrite the left-hand side as:
P(A ∩ B^c) = P(A) - P(A ∩ B) [Using the definition of complement: B^c represents the complement of event B]
4. Now, let's analyze the right-hand side:
P(A ∪ B) - P(B)
= (P(A ∩ B^c) + P(C)) - P(B) [Using the properties of set operations: A ∪ B = (A ∩ B^c) ∪ C]
= P(A ∩ B^c) + P(C) - P(B)
5. Now, we need to show that P(A ∩ B^c) + P(C) - P(B) equals P(A) - P(A ∩ B):
P(A ∩ B^c) + P(C) - P(B) = P(A) - P(A ∩ B)
6. Simplify the expression:
P(A ∩ B^c) + P(C) - P(B)
= P(A) - P(A ∩ B) [Using the definition of complement: B^c = A ∩ B^c]
7. Therefore, we have shown that:
P(A ∩ B^c) = P(A ∪ B) - P(B)
Now, let's move on to proving the second equality:
P(A ∩ B^c) = P(A) - P(A ∩ B)
This equality can be proven using basic set operations and the definition of probability.
Proof:
1. Start with the assumption that A and B are two events.
2. Suppose we define a new event, C, which represents the intersection of A and B.
3. By using the properties of set operations, we can rewrite the left-hand side as:
P(A ∩ B^c) = P(A) - P(A ∩ B) [Using the complement property: A ∩ B^c = A - A ∩ B]
4. Now, let's analyze the right-hand side:
P(A) - P(A ∩ B)
= P(A) - (P(A ∩ B) / P(B)) * P(B) [Using the definition of conditional probability]
= P(A) - P(A ∩ B) [Since P(A ∩ B) / P(B) = 1, as B is the sample space for P(A ∩ B)]
5. Therefore, we have shown that:
P(A ∩ B^c) = P(A) - P(A ∩ B)
In summary, both equalities have been proven using the properties of set operations and the definition of probability.