Question

Suppose a die is rolled six times and you need to find

a) The probability that at least two 4 come up

b) The probability that at least five 4's come up

Solve using the Binomial probability formula.

Answer 1

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is executed
p = success probability
mean = 6 * 0.1666
= 0.9996
II.
variance = npq
where
n = total number of repetitions experiment is executed
p = success probability
q = failure probability
variance = 6 * 0.1666 * 0.8334
= 0.833067
III.
standard deviation = sqrt( variance ) = sqrt(0.833067)
=0.912725

a) The probability that at least two 4 come up
P( X < 2) = P(X=1) + P(X=0)
= ( 6 1 ) * 0.1666^1 * ( 1- 0.1666 ) ^5 + ( 6 0 ) * 0.1666^0 * ( 1- 0.1666 ) ^6
= 0.7369
P( X > = 2 ) = 1 - P( X < 2) = 0.2631

b) The probability that at least five 4's come up
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 6 4 ) * 0.1666^4 * ( 1- 0.1666 ) ^2 + ( 6 3 ) * 0.1666^3 * ( 1- 0.1666 ) ^3 + ( 6 2 ) * 0.1666^2 * ( 1- 0.1666 ) ^4 + ( 6 1 ) * 0.1666^1 * ( 1- 0.1666 ) ^5 + ( 6 0 ) * 0.1666^0 * ( 1- 0.1666 ) ^6   
= 0.9993
P( X > = 5 ) = 1 - P( X < 5) = 0.0007