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WB JEE 2020

MCQ (More than One Correct Answer)
English
Bengali
A and B are independent events. The probability that both A and B occur is $${1 \over {20}}$$ and the probability that neither of them occurs is $${3 \over {5}}$$. The probability of occurrence of A is
A
$${1 \over {2}}$$
B
$${1 \over {10}}$$
C
$${1 \over {4}}$$
D
$${1 \over {5}}$$

Explanation

Given, $$P(A' \cap B') = {3 \over 5}$$

$$ \Rightarrow 1 - P(A \cup B) = {3 \over 5}$$

$$ \Rightarrow P(A \cup B) = {2 \over 5}$$

$$ \Rightarrow P(A) + P(B) - P(A \cap B) = {2 \over 5}$$

$$ \Rightarrow P(A) + P(B) - P(A)P(B) = {2 \over 5}$$ ...(i)

(since A and B are independent events)

Also, given $$P(A \cap B) = {1 \over {20}}$$

$$ \Rightarrow P(A)P(B) = {1 \over {20}}$$ ... (ii)

From Eqs. (i) and (ii), we get

$$P(A) = {1 \over 4}\,or\,{1 \over 5}$$

A ও B দুটি ঘটনা পরস্পর নির্ভরশীল নয়। A ও B উভয়ের একসঙ্গে ঘটার সম্ভাবনা $${1 \over {20}}$$ এবং কোনটি না ঘটার সম্ভাবনা $${3 \over 5}$$। A ঘটার সম্ভাবনা হবে

A
$${1 \over 2}$$
B
$${1 \over {10}}$$
C
$${1 \over 4}$$
D
$${1 \over 5}$$

Explanation

$$P(A \cap B) = {1 \over {20}} \Rightarrow P(A).P(B) = {1 \over {20}}$$

$$P(A \cup B) = 1 - {3 \over 5} = {2 \over 5}$$

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

বা, $${2 \over 5} = P(A) + {1 \over {20P(A)}} - {1 \over {20}}$$

বা, $${9 \over {20}} = P(A) + {1 \over {20P(A)}}$$

বা, $$20{(P(A))^2} - 9P(A) + 1 = 0$$

$$\therefore$$ $$(5P(A) - 1)(4P(A) - 1) = 0$$

$$\therefore$$ $$P(A) = {1 \over 5},{1 \over 4}$$

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