Answer to Question #335889 in Statistics and Probability for akeel

Question

In a batch of 15 jars of instant coffee, 5 have been under filled. Suppose 6 of these jars are selected without replacement. If more than 1 of these selected jars are under filled, the whole batch is sent back for refill. What is the probability of the batch being sent back for refill? (Round off to 2 decimal places.)

Answer 1

Suppose that $X$ is a random variable and it denotes a number of selected under filled jars. The aim is to find the probability $P(X>1)$. $P(X>1)=1-P(X=0)-P(X=1)$. There are $C_{15}^6=\frac{15!}{9!6!}=\frac{10\cdot11\cdot12\cdot13\cdot14\cdot15}{6!}=5005$ different ways to choose $6$ jars from $15$. There are $C_{10}^6=\frac{10!}{4!6!}=\frac{7\cdot8\cdot9\cdot10}{4!}=210$ ways to choose $6$ filled jars from $10$. There are $C_{10}^5=\frac{10!}{5!5!}=\frac{6\cdot7\cdot8\cdot9\cdot10}{5!}=252$ ways to get $5$ filled jars from $10$. There are $5$ ways to select $1$ under filled jar from $5$. $P(X=0)=\frac{C_{10}^6}{C_{15}^6}=\frac{210}{5005}\approx0.042$. $P(X=1)=\frac{5\cdot C_{10}^5}{C_{15}^6}=\frac{5\cdot210}{5005}\approx0.210$. Thus, the probability that the batch will be sent back is: $1-0.042-0.210\approx0.748$.

Answer: the probability that the batch will be sent back is: $0.75$ (it is rounded to $2$ decimal places).