Answer: 0.2962
Step-by-step 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. [tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]
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 , [tex]P(8\leq x\leq 10)=P(x=8)+P(x=9)+P(x=10)[/tex]
[tex]=^{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}\\\\=\dfrac{15!}{8!7!}(0.75)^8(0.25)^{7}+\dfrac{15!}{9!6!}(0.75)^9(0.25)^{6}+\dfrac{15!}{10!5!}(0.75)^{10}(0.25)^{5}\\\\=0.0393204716966+0.091747767292+0.165145981126\\\\=0.296214220114\approx0.2962[/tex] [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
