Question

1. From past experience it is known that 90% of one-year old children can distinguish their mother’s voice from the voice of a similar sounding female. A random sample of 50 one-year olds is given this voice recognition test.
   1. Find the probability that exactly 3 children do not recognize their mother’s voice. (2 points)
2. 2. Find the probability that 3 or fewer children do not recognize their mother’s voice. (2 points)
3. 3. What is the probability that at least 47 children recognize their mother’s voice?(2 points)
4. 4. Find the probability that exactly 90% of the 50 children do recognize their mother’s voice. (2 points)
5. 5. What is the probability that exactly 49 of the 50 children recognize their mother’s voice? (2 points)
6. 6. Find the probability all 50 children recognize their mother’s voice. (2 points)
7. 7. Let the random variable x denote the number of children who recognize their mother’s voice. Find the mean of x. (2 points)
8. 8. Let the random variable x denote the number of children who do not recognize their mother’s voice. Find the mean of x. (2 points)
   8. Let the random variable x denote the number of children who do not recognize their mother’s voice. Find the variance of x. (2 points)

Answer 1

Here from the explaination given in the question it can be observed that binomial distribution can be used to find the probabilities where

and   

where P is probability that a one year old child can distinguish his/her mother's voice correctly.

n=50

a) To find probability that 3 children does not recognise there mother's voice correctly that is 47 identify it correctly.

so probability is

b) probability that 3 or fewer children does not recognise voice correctly is equal to 47,48,49,50 children answer it correctly, so probability is

This is equal to =

Hence probability is 0.250292

c) To find probability that atleast 47 children recognise their mother's voice is equal to sum of probabilites for 47,48,49,50 children who recognised their mother's voice correctly

This is same as part b because 3 or fewer children not recognising their mother's voice is equal to 47,48,49,50 recognising it correctly.

Hence answer is 0.250292

d) Firstly 90% of 50 is 45, so to find probability that exactly 45 children recognised voice correctly is given by

Hence probability is 0.1849246