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  • Let A and B be two independent events. The probability that both A and B occur is 1/30 and the probability that neither A nor B occurs is 2/3 The respective probabilities of A and B are

Let A and B be two independent events. The probability that both A and B occur is 1/30 and the probability that neither A nor B occurs is 2/3 The respective probabilities of A and B are

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$ Let P(A)=x and P(B)=y \\ $ \therefore x y=\frac{1}{30}...(i) $ and $ (1-x)(1-y)= \frac{2}{3 } \\ 1-x-y+x y=\frac{2}{3} \\\\ x+y= 1- \frac{2}{3} + \frac{1}{30} \\\\ x+y= \frac{30-20+1}{30} \\\\ x+y= \frac{11}{30}...(ii) \\ \\ $ from equation (i) and Equation (ii) \\ $ x(\frac{11}{30} -x)= \frac{1}{30} \\\\ 11x - 30 x^2 =1 \\ 30x^2 -11x+1 = 0 \\ x = \frac{11 \pm \sqrt{(11)^2 - 4 \times 30 \times 1}}{2 \times 30} \\\\ x = \frac{11 \pm \sqrt{121 - 120}}{60} \\\\ x = \frac{6}{30}\ or\ x = \frac{5}{30} \\\\ y = \frac{5}{50} \ or\ y =\frac{6}{30}\\\\ $ Hence x and y can take values as 0.2 and 0.1666 or vice-versa.

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Ravindra Pindel

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