Question

Suppose that 75% of all trucks undergoing a brake inspection at a certain inspection facility pass the inspection. Consider groups of 15 trucks and let X be the number of trucks in a group that have passed the inspection. What is the probability that there will be between 8 and 10 trucks (inclusive) which pass the inspection?

Answer 1

Answer: 0.2962

Explanation:

Given : The proportion of all trucks undergoing a brake inspection at a certain inspection facility pass the inspection = 0.75

Since , the proportion of trucks pass the inspection is certain for each ruck.

⇒ It is Binomial distributed.

i.e.
`P(x)=^nC_xp^x(1-p)^(n-x)`

We consider : Groups of 15 trucks and Let X be the number of trucks in a group that have passed the inspection.

i.e. n= 15

Then ,
`P(8\leq x\leq 10)=P(x=8)+P(x=9)+P(x=10)`

`=^(15)C_(8)(0.75)^8(0.25)^(7)+^(15)C_(9))(0.75)^9(0.25)^(6)+^(15)C_(10)(0.75)^(10)(0.25)^(5)\\\\=(15!)/(8!7!)(0.75)^8(0.25)^(7)+(15!)/(9!6!)(0.75)^9(0.25)^(6)+(15!)/(10!5!)(0.75)^(10)(0.25)^(5)\\\\=0.0393204716966+0.091747767292+0.165145981126\\\\=0.296214220114\approx0.2962` [Rounded to nearest 4 decimal places.]

Hence, the probability that there will be between 8 and 10 trucks (inclusive) which pass the inspection =0.2962