A manufacturer of ceramic blades estimates that 28 of all bl

A manufacturer of ceramic blades estimates that 2.8% of all blades produced
are too brittle to use. Suppose we take a random sample of 400 blades and test them for
brittleness. What is the probability that at least 8 blades but no more than 10 blades will
be too brittle to use? Find the exact probability; do not use an approximation here.

Solution

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 = 8 ) = ( 400 8 ) * ( 0.028^8) * ( 1 - 0.028 )^392
= 0.0837
P( X = 9 ) = ( 400 9 ) * ( 0.028^9) * ( 1 - 0.028 )^391
= 0.105
P( X = 10 ) = ( 400 10 ) * ( 0.028^10) * ( 1 - 0.028 )^390
= 0.1183

P( 8 <= X <= 10) = 0.0837 + 0.105 + 0.1183 = 0.307