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Let $A$ and $B$ be any two arbitrary events, then, which one of the following is TRUE?

  1. $P (A \cap B) = P(A)P(B)$
  2. $P (A \cup B) = P(A)+P(B)$
  3. $P (A \mid B) = P(A \cap B)P(B)$
  4. $P (A \cup B) \leq P(A) + P(B)$
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6 Answers

Best answer
53 votes
53 votes
  1. Is true only if events are independent.
  2. Is true only if events are mutually exclusive i.e. $P (A \cap B) = 0$
  3. Is false everywhere.
  4. Is always true as $P (A \cup B) = P(A)+P(B)-P (A \cap B)$

Since, $P (A \cap B) >=0$, $P (A \cup B) \leq P(A)+P(B)$

Correct Answer: D.

edited by
9 votes
9 votes
If A and B are mutually exclusive events then:

A$\cap$B = $\Phi$ and A$\cup$B = A+B

If A and B are non mutually exclusive events then:

A$\cap$B != $\Phi$ and A$\cup$B = A+B-A$\cap$B (principle of inclusion-exclusion)

therefore A$\cup$B <= A + B

so D is the answer...
edited by
7 votes
7 votes
Option D is Ans

If A and B be two arbitrary events, then

P (A ∪ B) = P (A) +P (B) - P(A $\cap$ B)

Let say 10=7+8-5

So now  10 <= 7+8

So option D ....P (A ∪ B) <= P (A) +P (B) is correct Ans.
Answer:

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