A fair die is rolled 8 times. What is the probability that the die comes up 6 exactly twice?

Solution:

Given A fair die is rolled 8 times.

Let P be the probability of getting 6 exactly twice

The probability (p) of one 6 in first roll is (1/6),

The probability (q) of not getting 6 in first roll is 5/6

So, getting 6 exactly twice in 8 rolls can be given by binomial distribution as

P = ⁿC_r p^r. q^{n - r}

Here n = 8 and r = 2

P(r=2) = ⁸C₂(1/6)².(5/6)^{8 - 2}

P = ⁸C₂(1/6)².(5/6)⁶

P = 8!/ (8 - 2)!2! (1/36) (5/6)⁶

P = 8(7)/2 (1/36) (5/6)⁶

P = 28/36 (5/6)⁶

P = (7/9)(5/6)⁶

P = (7/9)(15625/46656)

P = 109375/ 419904

P = 0.026

P = 1/4

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A fair die is rolled 8 times. What is the probability that the die comes up 6 exactly twice?

Summary:

A fair die is rolled 8 times, the probability that the die comes up 6 exactly twice is P = 1/4