TS EAMCET · Maths · Probability
If \(\mathrm{A}\) and \(\mathrm{B}\) are two events of a random experiment such that \(P(A \cup B)=P(A \cap B)\), then which one amongst the following four options in not true?
- A \(A\) and \(B\) are equally likely
- B \(\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right)=0\)
- C \(P\left(A^{\prime} \cap B\right)=0\)
- D \(\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=1\)
Answer & Solution
Correct Answer
(D) \(\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=1\)
Step-by-step Solution
Detailed explanation
\[ \begin{array}{ll} & P(A \cup B)=P(A \cap B) \\ \because \quad & P(A \cup B)=P(A)+P(B)-P(A \cap B) \\ \therefore \quad & 2 P(A \cap B)=2 P(A \cup B)=P(A)+P(B) \end{array} \] Clearly \(P(A)+P(B)=2 P(A \cap B)\) which can't be always 1 . \(\therefore \quad P(A)+P(B)=1\) is wrong.
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