Duffy Lighting Inc products LED light bulbs Charlie Huff is

Duffy Lighting, Inc. products LED light bulbs. Charlie Huff is the quality control manager for Duffy Lighting and he has his department regularly check the light bulbs they manufacture.  Historically, the failure rate for their LED lightbulbs was 10.0%. Charlie select a random sample of 30 light bulbs from a current production run.

Out of those 30, what is the probability that:

a. none are defective?

b. exactly one is defective?

c. 4 or fewer are defective?

d. 3 or more are defective?

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

a)
P( X = 0 ) = ( 30 0 ) * ( 0.1^0) * ( 1 - 0.1 )^30
= 0.0424
b)
P( X = 1 ) = ( 30 1 ) * ( 0.1^1) * ( 1 - 0.1 )^29
= 0.1413

c)
P( X < = 4) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 30 4 ) * 0.1^4 * ( 1- 0.1 ) ^26 + ( 30 3 ) * 0.1^3 * ( 1- 0.1 ) ^27 + ( 30 2 ) * 0.1^2 * ( 1- 0.1 ) ^28 + ( 30 1 ) * 0.1^1 * ( 1- 0.1 ) ^29 + ( 30 0 ) * 0.1^0 * ( 1- 0.1 ) ^30
= 0.8245

d)
P( X < 3) = P(X=2) + P(X=1) + P(X=0)
= ( 30 2 ) * 0.1^2 * ( 1- 0.1 ) ^28 + ( 30 1 ) * 0.1^1 * ( 1- 0.1 ) ^29 + ( 30 0 ) * 0.1^0 * ( 1- 0.1 ) ^30
= 0.4114

P( X > = 3 ) = 1 - P( X < 3) = 0.5886