Consider two events E₁ and E₂ such that P(E₁) = 1/2, P(E₂) = 1/3, and P(E₁ ∩…
1999
Consider two events E₁ and E₂ such that P(E₁) = 1/2, P(E₂) = 1/3, and P(E₁ ∩ E₂) = 1/5. Which of the following statements is true?
- A.
P(E₁ ∩ E₂) = 2/3
- B.
Events E₁ and E₂ are independent.
- C.
Events E₁ and E₂ are not independent.
- D.
P(E₁ | E₂) = 4/5
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Correct answer: C
For two events to be independent, P(E₁ ∩ E₂) must be equal to P(E₁)P(E₂). Here, P(E₁)P(E₂) = (1/2)(1/3) = 1/6, while P(E₁ ∩ E₂) = 1/5. Since 1/5 ≠ 1/6, E₁ and E₂ are not independent. Also, P(E₁ | E₂) = P(E₁ ∩ E₂)/P(E₂) = (1/5)/(1/3) = 3/5, so the statement P(E₁ | E₂) = 4/5 is false. Therefore, the true statement is: Events E₁ and E₂ are not independent.