A die is to be tossed 144 times and the number of time a 6 a

A die is to be tossed 144 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 28 6\'s appear? Give your answer to two decimal places, without the leading zero.

d) What is the approximate probability that more than 20 but fewer than 28 6\'s appear? Give your answer to two decimal places, without the leading zero.

Solution

a) What is the mean number of 6\'s expected? Give your answer to the nearest integer.

mean=n*p=144*(1/6) =24

-------------------------------------------------------------------------------------------------------------------------

b) What is the variance of the number of 6\'s expected? Give your answer to the nearest integer.

variance= n*p*(1-p) =144*(1/6)*(5/6) =20

-------------------------------------------------------------------------------------------------------------------------

c) What is the approximate probability that fewer than 28 6\'s appear? Give your answer to two decimal places, without the leading zero.

P(X<28) = P((X-mean)/s <(28-24)/sqrt(20))

=P(Z<0.89) =0.8133 (from standard normal table)

-------------------------------------------------------------------------------------------------------------------------

d) What is the approximate probability that more than 20 but fewer than 28 6\'s appear? Give your answer to two decimal places, without the leading zero.

P(20<X<28) = P((20-24)/sqrt(20)<Z<(28-24)/sqrt(20))

=P(-0.89<Z<0.89) =0.6265 (from standard normal table)