If an unbiased die, marked with ?2, ?1, 0, 1, 2, 3 on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is [2023]

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Solution

Q.
If an unbiased die, marked with −2, −1, 0, 1, 2, 3 on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is [2023]

1 $\frac{440}{2592}$

2 $\frac{881}{2592}$

3 $\frac{27}{288}$

4 $\frac{521}{2592}$

Ans.
(4)

The markings on an unbiased die are −2, −1, 0, 1, 2, 3.

$P (positive number) = \frac{3}{6} = \frac{1}{2}$

$P (negative number) = \frac{2}{6} = \frac{1}{3}$

Let $A =$ Product is positive.

$A = (5 positive numbers) or (1 positive, 4 negative) or (3 positive, 2 negative)$

$P (A) = {}^{5}C_{5} {(\frac{1}{2})}^{5} + {}^{5}C_{1} {(\frac{1}{3})}^{4} {(\frac{1}{2})}^{1} + {}^{5}C_{3} {(\frac{1}{2})}^{3} \times {(\frac{1}{3})}^{2}$

$= \frac{1}{32} + \frac{5}{162} + \frac{10}{72} = \frac{521}{2592}$

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