In: Statistics and Probability
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