Let π΄ and π΅ be two events in a probability space with π(π΄) = 0.3, π(π΅) =β¦
2024
Let π΄ and π΅ be two events in a probability space with π(π΄) = 0.3, π(π΅) = 0.5, and π(π΄ β© π΅) = 0.1. Which of the following statements is/are TRUE?
- A.
The two events π΄ and π΅ are independent
- B.
π(π΄ βͺ π΅) = 0.7
- C.
\(P(A \cap B^c) \)Β = 0.2, whereΒ\(B^c\)is the complement of the event B - D.
\(P(A^c \cap B^c) \)Β = 0.4, whereΒ\(A^c\)andΒ\(B^c\)are the complements of the events π΄ and π΅, respectively
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Correct answer: B, C
Summary: The true statements are P(A βͺ B) = 0.7 and P(A β© B^c) = 0.2. The statements about independence and P(A^c β© B^c) = 0.4 are false.
Check independence: compute P(A)P(B) = 0.3 Γ 0.5 = 0.15, which is not equal to the given P(A β© B) = 0.1. Therefore the events are not independent.
Compute the union: P(A βͺ B) = P(A) + P(B) β P(A β© B) = 0.3 + 0.5 β 0.1 = 0.7. So the union statement is correct.
Compute P(A β© B^c): since A = (A β© B) βͺ (A β© B^c) with disjoint parts, P(A β© B^c) = P(A) β P(A β© B) = 0.3 β 0.1 = 0.2. So this statement is correct.
Compute P(A^c β© B^c): this equals 1 β P(A βͺ B). Using P(A βͺ B) = 0.7 gives P(A^c β© B^c) = 1 β 0.7 = 0.3, so the claim of 0.4 is incorrect.
Final correct statements: P(A βͺ B) = 0.7 and P(A β© B^c) = 0.2.
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