JEE Main 2021 · 24 Feb · Shift 1 · Mathematics Q82

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MathematicsInteger+4 / -1Question 82 of 90

Let B $_i$ (i = 1, 2, 3) be three independent events in a sample space. The probability that only B $_1$ occur is α, only B $_2$ occurs is β and only B $_3$ occurs is γ. Let p be the probability that none of the events B $_i$ occurs and these 4 probabilities satisfy the equations (α − 2β) p = αβ and (β − 3γ) p = 2βγ (All the probabilities are assumed to lie in the interval (0, 1)). Then $\dfrac{P(B_1)}{P(B_3)}$ is equal to _______.

Your answer

Answer and worked solution

Official answer: 6

Odds. With $x_i=P(B_i)$ , $\frac\alpha p=\frac{x_1}{1-x_1}=a$ , $\frac\beta p=b$ , $\frac\gamma p=c$ . The equations become $a-2b=ab$ and $b-3c=2bc$ , i.e. $\frac1b-\frac2a=1$ and $\frac1c-\frac3b=2$ .

Ratio. With $u=\frac1a$ : $\frac1b=2u+1$ , $\frac1c=6u+5$ . Since $x_i=\frac1{1+1/a_i}$ , $\dfrac{x_1}{x_3}=\dfrac{1+\frac1c}{1+u}=\dfrac{6u+6}{u+1}=6$ .

Answer: 6

Topics:Independent events

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JEE Main 2021 · 24 Feb · Shift 1, Mathematics question 82