In a university elevator there is a sign indeicating a 16per

In a university elevator there is a sign indeicating a 16-person limit as well as a weight limit of 2500 lbs. suppose that the weight of students, faculty, and staff is approximately Normally distributed with a mean weight of 150 lbs and a standard deviation of 27 lbs.

What is the probability that exactly 4 groups will exceed the maximum weight limit

Solution

Normal Distribution
Mean ( u ) =150
Standard Deviation ( sd )=27
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
P(X > 156.25) = (156.25-150)/27
= 6.25/27 = 0.2315
= P ( Z >0.231) From Standard Normal Table
= 0.4085                  

Probability of exceeding the limit is 0.4085

It follows binomial with B ~ ( 12,0.4085)
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

P( X = 4 ) = ( 12 4 ) * ( 0.4085^4) * ( 1 - 0.4085 )^8
= 0.2065