A fair die is rolled 8 times. What is the probability that the die comes up 6 exactly twice? What is the probability that the die comes up an odd number exactly five times? Find the mean number of times a 6 comes up. Find the mean number of times an odd number comes up. Find the standard deviation of the number of times a 6 comes up. Find the standard deviation of the number of times an odd number comes up.

0.2605, 0.2188, 1.33, 4, 1.0540, 1.4142

Step-by-step explanation:

A fair die is rolled 8 times.  

a. What is the probability that the die comes up 6 exactly twice?  

b. What is the probability that the die comes up an odd number exactly five times?  

c. Find the mean number of times a 6 comes up.  

d. Find the mean number of times an odd number comes up.  

e. Find the standard deviation of the number of times a 6 comes up.  

f. Find the standard deviation of the number of times an odd number comes up.

a. A die is rolled 8 times. If A represent the number of times a 6 comes up. For a fair die the probability that the die comes up 6 is 1/6 - Thus A ~ Bin(8, 1/6)

The probability mass function  of the random variable A is  

hence, p(6 twice) implies P(A=2)

that is P(2) substitute A = 2

p(2) = 0.2605  

b. If B represent the number of times an odd number comes up. For the fair die the probability that an odd number comes up is 0.5.

Thus B ~ Bin(8, 1/2 )

The probability mass function of the random variable B is given by

hence p(odd comes up 5 times) is

p(5) = 0.2188

c. let the mean no of times a 6 comes up be μₐ

   and let the total number of outcomes be n

   using the formula μₐ = nρₐ

   μₐ = 8 * 1/6

        = 1.33

d. let the mean nos of times an odd nos comes up be μₓ

   let the total outcomes be n = 8

   let the probability odd be pb = 1/2

   μₓ = npb

        = 8 * (1/2)

        = 4

e. the standard deviation of a random variable A is given as follows

σₐ

where p = 1/6 (prob 6 outcome)

n = total outcomes = 8

  = 1.0540

f. the standard dev of the binomial random variable Y is given by

σ

where p = 1/2 and n = 8

  =  

  = 1.4142

this is answer of 144 cubic meters