Given F and G are two events of an experiment with P(F) = 0.17, P(G) = 0.49, and P(FUG) = 0.54, calculate P((G^C ∩ F)^C).
Understand the Problem
The question is asking to calculate the probability of the complement of the intersection of the complement of event G and event F, given the probabilities of events F, G, and the union of F and G. We are given:
- P(F) = 0.17
- P(G) = 0.49
- P(F U G) = 0.54
We need to find P((G^C ∩ F)^C).
Answer
$0.95$
Answer for screen readers
$P((G^C \cap F)^C) = 0.95$
Steps to Solve
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Find $P(F \cup G)$ using the inclusion-exclusion principle Find $P(F \cap G)$ using the formula: $P(F \cup G) = P(F) + P(G) - P(F \cap G)$
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Plug in the given values $0.54 = 0.17 + 0.49 - P(F \cap G)$
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Solve for $P(F \cap G)$ $P(F \cap G) = 0.17 + 0.49 - 0.54 = 0.66 - 0.54 = 0.12$
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Find $P(G^C \cap F)$ This represents the probability of F occurring but not G. $P(G^C \cap F) = P(F) - P(F \cap G) = 0.17 - 0.12 = 0.05$
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Find $P((G^C \cap F)^C)$ This is the complement of $P(G^C \cap F)$. $P((G^C \cap F)^C) = 1 - P(G^C \cap F) = 1 - 0.05 = 0.95$
$P((G^C \cap F)^C) = 0.95$
More Information
The probability of the complement of the intersection of the complement of event G and event F is 0.95. This means that there's a high chance that either G occurs or F does not occur.
Tips
A common mistake is to misinterpret the notation and incorrectly calculate the intersection or complement probabilities. Another mistake is to apply the inclusion-exclusion principle incorrectly. Careful attention to the definitions and formulas is crucial.
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