A die is to be tossed 108 times and the number of time a 6 a

A die is to be tossed 108 times and the number of time a 6 appears is to be recorded. a) What is the mean number of 6?s expected? Give your answer to the nearest integer. b) What is the variance of the number of 6?s expected? Give your answer to the nearest integer c) What is the approximate probability that fewer than 20 6\'s appear? Give your answer to two decimal places, without the leading zero. d) What is the approximate probability that more than 15 but fewer than 20 6\'s appear? Give your answer to two decimal places, without the leading zero.

Solution

probability 6 appears in a single toss, p = 1/6

n = 108

a)

mean number of 6\'s expected = n*p = 108*(1/6) = 18

b)

Variance = np*(1-p) = 108*(1/6)*(5/6) = 15

c)

using normal approximation of binomial,

mu = 18, sigma= sqrt(variance) = sqrt(15) = 3.87

z-score at 20 = (20-18)/3.87 = 0.5167

P( fewer than 20 6\'s appear) = P( z < 0.5167) = 0.69732

d)

z-score at 15 = (15-18)/3.87 = -0.7751

P( 15 < x <20 ) = P( -0.7751 < z < 0.5167) = 0.69732 - 0.21914 = 0.4781