Question

On the average,1% of a specific engine componentis found to not perform to standards when tested at the engine assembly plant. For each shipment of 100 components, they are all tested and if more than two are found to be non-performing the entireshipment is returned. What is the probability that a shipment is returned?

Answer 1

Answer: 0.07938

Explanation:

Let X be a binomial variable that represents the components of a specific engine.

As per given , the probability of componentis found to not perform to standards : p= 1%=0.01

Binomial probability formula :

`P(X=x)=\ ^nC_xp^x(1-p)^(n-x)`

Sample size : n= 100

If more than two are found to be non-performing the entire shipment is returned.

Now, Required probability :
`P(X>2)=1-P(X\leq 2)`

`=1-(P(X=0)+P(X=1)+P(X=2))\\\\=1- (^(100)C_(0)(0.01)^0(0.99)^(100)+^(100)C_(1)(0.01)^(1)(0.99)^(99)+^(100)C_(2)(0.01)^(2)(0.99)^(98))\\\\=1-((0.99)^(100)+(100)(0.01)(0.99)^(99)+(100!)/(2!98!)(0.01)(0.99)^98)\\\\=1-(0.36603+0.36973+0.18486)\\\\=1-(0.92062)\\\\=0.07938`

So, the probability that a shipment is returned = 0.07938