Let A and B be the events such that P(A) = 1/2, P(B) = 7/12 and P(not

1.	Let A and B be the events such that P(A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4.State whether A and B are(i) mutually exclusive(ii) independent
Answer» Given: A and B are the events such that P(A) = \(\frac{1}{2}\) and P(B) = \(\frac{7}{12}\) and P(not A or not B) = \(\frac{1}{4}\) To Find: (i) If A and B are mutually exclusive Since P(not A or not B) = \(\frac{1}{4}\) i.e., P\((\overline A\cup\overline B)\) = \(\frac{1}{4}\) we know that , P\((\overline A\cup\overline B)\)) = P(A ∩ B)' = 1 - P(A ∩ B) = 0 But here P(A ∩ B) \(\neq\) 0 Therefore , A and B are not mutually exclusive. (ii) If A and B are independent The condition for two events to be independent is given by P(E₁ ∩ E₂) = P(E₁) x P(E₂) = \(\frac{1}{2}\times \frac{7}{12}\) = \(\frac{7}{24}\) Equation 2 Since Equation 1 \(\neq\) Equation 2 \(\Rightarrow\) A and B are not independent

Let A and B be the events such that P(A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4.State whether A and B are(i) mutually exclusive(ii) independent

Answer»

Given: A and B are the events such that P(A) = \(\frac{1}{2}\) and P(B) = \(\frac{7}{12}\) and

P(not A or not B) = \(\frac{1}{4}\)

To Find:

(i) If A and B are mutually exclusive

Since P(not A or not B) = \(\frac{1}{4}\) i.e., P\((\overline A\cup\overline B)\) = \(\frac{1}{4}\)

we know that , P\((\overline A\cup\overline B)\)) = P(A ∩ B)' = 1 - P(A ∩ B) = 0

But here P(A ∩ B) \(\neq\) 0

Therefore , A and B are not mutually exclusive.

(ii) If A and B are independent

The condition for two events to be independent is given by

P(E₁ ∩ E₂) = P(E₁) x P(E₂)

= \(\frac{1}{2}\times \frac{7}{12}\)

= \(\frac{7}{24}\) Equation 2

Since Equation 1 \(\neq\) Equation 2

\(\Rightarrow\) A and B are not independent

Discussion

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