Answer to Question #122517 in Statistics and Probability for nasar

Question

Super bikes arrive in a 12 lot showroom. Inspections are carried out on six out of every 12. For
one lot, it is known 2 out of 12 do not meet the safety standards prescribed. What is the
probability that at least 2 out of the 6 that were tested from that lot will not meet safety
standards?

Answer 1

What is the probability that 0 out of the 6 that were tested from that lot will not meet safety standards?

$$\dfrac{\dbinom{2}{0}\dbinom{12-2}{6-0}}{\dbinom{12}{6}}=\dfrac{1\cdot\dfrac{10!}{6!(10-6)!}}{\dfrac{12!}{6!(12-6)!}}=$$

$$=\dfrac{6(5)}{12(11)}=\dfrac{5}{22}$$

What is the probability that 1 out of the 6 that were tested from that lot will not meet safety standards?

$$\dfrac{\dbinom{2}{1}\dbinom{12-2}{6-1}}{\dbinom{12}{6}}=\dfrac{2\cdot\dfrac{10!}{5!(10-5)!}}{\dfrac{12!}{6!(12-6)!}}=$$

$$=\dfrac{2(6)(6)}{12(11)}=\dfrac{6}{11}$$

What is the probability that at least 2 out of the 6 that were tested from that lot will not meet safety standards?

$$1-\dfrac{5}{22}-\dfrac{6}{11}=\dfrac{5}{22}\approx0.227273$$