Please show work THANK YOU From past experience it is known

Please show work!! THANK YOU

From past experience, it is known 90% of 1-year-old children can distinguish their mother\'s voice from the voice of a similar sounding female. A random sample of 20 1-year-olds is given this voice recognition test.

This is a binomial distibution

n=                            p=                               q=

I know the value of p will change depending on the children recognizing or not recognizing the voice

A. find the probability all 20 children recognize their mothers voice.

B. Find the probability at least 3 children DO NOT RECONIZE their mothers voice

C. find the mean and standard deviation of x, where x is the random variable denoting the number of children who recognize their mother\'s voice.

D. Find the probability that at most 4 children RECOGNIZE their mothers voice

Solution

Binomial 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

n = 20
p = 0.90
q = 0.10
a)
P( X = 20 ) = ( 20 20 ) * ( 0.9^20) * ( 1 - 0.9 )^0
= 0.1216

b)
P( 3 don\'t recognize their voice) = P( 17 recognized their voice)
P( X = 17 ) = ( 20 17 ) * ( 0.9^17) * ( 1 - 0.9 )^3
= 0.1901

c)
Mean ( np ) =18
Standard Deviation ( npq )= 20*0.9*0.1 = 1.3416

d)
P( X < = 4) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 20 4 ) * 0.1^4 * ( 1- 0.1 ) ^16 + ( 20 3 ) * 0.1^3 * ( 1- 0.1 ) ^17 + ( 20 2 ) * 0.1^2 * ( 1- 0.1 ) ^18 + ( 20 1 ) * 0.1^1 * ( 1- 0.1 ) ^19 + ( 20 0 ) * 0.1^0 * ( 1- 0.1 ) ^20
= 0.9568