Show that if A and B are independent events P(A or B)-1 -[1-P(A)][1-P(B)].
To prove this, we can start by using the definition of probability for the union of two events:
P(A∪B) = P(A) + P(B) - P(A∩B)
Since we are given that events A and B are independent, we know that P(A∩B) = P(A) * P(B). Substituting this into the above equation, we get:
P(A∪B) = P(A) + P(B) - P(A) * P(B)
Now, let's simplify the expression on the right-hand side:
P(A∪B) = P(A) + P(B) - P(A) * P(B)
= 1 - (1 - P(A)) * (1 - P(B))
This is the expression we need to prove. Now, let's break down the steps of how we arrived at this expression:
1. Start with the definition of probability for the union of two events: P(A∪B) = P(A) + P(B) - P(A∩B).
2. Since A and B are independent events, P(A∩B) = P(A) * P(B).
3. Substitute P(A) * P(B) for P(A∩B) in the equation.
4. Simplify the expression to obtain: P(A∪B) = 1 - (1 - P(A)) * (1 - P(B)).
Therefore, we have shown that if events A and B are independent, then P(A∪B) = 1 - (1 - P(A)) * (1 - P(B)).