If A and B are two events such that P(A) = 1/4, P(B) = 1/2, and P(A ∩ B) = 1/8, find P(A' ∩ B').
Understand the Problem
The question asks to find the probability of the complement of A and B, given the probabilities of A, B, and their intersection. This involves using the formula P(A' ∩ B') = 1 - P(A ∪ B), and the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Answer
$P(A' \cap B') = 0.1$
Answer for screen readers
$P(A' \cap B') = 0.1$
Steps to Solve
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Find $P(A \cup B)$ We will use the formula for the probability of the union of two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Substitute the given values: $P(A \cup B) = 0.6 + 0.5 - 0.2 = 0.9$
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Find $P(A' \cap B')$ We know that $P(A' \cap B') = P((A \cup B)')$ (DeMorgan's Law). Also, $P((A \cup B)') = 1 - P(A \cup B)$. Substitute the value we found for $P(A \cup B)$: $P(A' \cap B') = 1 - 0.9 = 0.1$
$P(A' \cap B') = 0.1$
More Information
The probability of the complement of A and B occurring is 0.1. This means that there is a 10% chance that neither A nor B occurs.
Tips
A common mistake is to incorrectly apply DeMorgan's Law or to confuse the union and intersection of events. Another mistake is to forget to subtract $P(A \cap B)$ when calculating $P(A \cup B)$.
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