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Three students independently took an exam. $X$ be a random variable representing the number of students passed. Following is the probability distribution of $X .$
$X$ 0 1 2 3
$P (X)$ $\frac{2}{5}$ $\frac{13}{30}$ $\frac{3}{20}$ $\frac{1}{60}$

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Highest probability of an individual student to pass the exam, is $\frac{1}{2} .$

Highest probability of an individual student to pass the exam, is $\frac{1}{3} .$

Lowest probability of an individual student to pass the exam, is $\frac{1}{5} .$

Lowest probability of an individual student to pass the exam, is $\frac{1}{4} .$

answer is B, C.

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Detailed Solution

$P (X = 0) = 1 - S_{1} + S_{2} - S_{3} = \frac{2}{5}$

$P (X = 1) = S_{1} - 2 S_{2} + 3 S_{3} = \frac{13}{30}$

$P (X = 2) = S_{2} - 3 S_{3} = \frac{3}{20}$

$P (X = 3) = S_{3} = \frac{1}{60}$

Solving $S_{1} = \frac{47}{60}, S_{2} = \frac{1}{5}$
Hence we need roots of $x^{3} - \frac{47}{60} x^{2} + \frac{1}{5} x - \frac{1}{60} = 0$
Probabilities of individual students are $\frac{1}{3}, \frac{1}{4}, \frac{1}{5}$

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