Maths · Probability: Probability of an event, addition and multiplication theorems of probability
If and are independent events such that <\mathbf{1}, \mathbf{0}<\boldsymbol{P}(\
If \( A \) and \( B \) are independent events such that \( \mathbf{0}<\boldsymbol{P}(\boldsymbol{A})<\mathbf{1}, \mathbf{0}<\boldsymbol{P}(\boldsymbol{B})<\mathbf{1}, \) then This question has multiple correct options
- A. A, B are mutually exclusive
- B. A and \( \bar{B} \) are independent
- C. \( \bar{A} \) and \( \bar{B} \) are independent
- D. \( P(A / B)+P(\bar{A} / B)=1 \)
Step-by-step solution
Given A and B are independent with 0<P(A),P(B)<1. For independence, P(A∩B)=P(A)P(B). Then P(A∩B̅)=P(A)-P(A∩B)=P(A)-P(A)P(B)=P(A)(1-P(B))=P(A)P(B̅), so A and B̅ are independent. Similarly, A̅ and B̅ are independent, and P(A/B)+P(A̅/B)=P(A)+P(A̅)=1. Options B, C, D are correct; A is false because independent events are not mutually exclusive unless probability zero.
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