SATHEE: Probability 3 Question 41

Probability 3 Question 41

41. $A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$ . Find the probability of the occurrence of $A$ .

(1984, 2M)

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Answer:

Correct Answer: 41. $\frac{1}{3}$ or $\frac{1}{2}$

Solution:

Given, $P (A) \cdot P (B) = \frac{1}{6}, P (\bar{A}) \cdot P (\bar{B}) = \frac{1}{3}$

$∴ [1 - P (A)] [1 - P (B)] = \frac{1}{3}$

$\begin{aligned} Let P (A) = x and P (B) = y \\ \Rightarrow (1 - x) (1 - y) = \frac{1}{3} and x y = \frac{1}{6} \\ \Rightarrow 1 - x - y + x y = \frac{1}{3} and x y = \frac{1}{6} \\ \Rightarrow x + y = \frac{5}{6} and x y = \frac{1}{6} \\ \Rightarrow x \frac{5}{6} - x = \frac{1}{6} \\ \Rightarrow 6 x^{2} - 5 x + 1 = 0 \\ \Rightarrow (3 x - 1) (2 x - 1) = 0 \\ \Rightarrow x = \frac{1}{3} and \frac{1}{2} \\ ∴ P (A) = \frac{1}{3} or \frac{1}{2} \end{aligned}$

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Probability 3 Question 41

41. A and B are two independent events. The probability that both A and B occur is 16 and the probability that neither of them occurs is 13. Find the probability of the occurrence of A.

Answer:

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41. $A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$ . Find the probability of the occurrence of $A$ .