Answer to Question #335869 in Statistics and Probability for sdu

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?

Answer 1

Let $X=$ the number of  jars under filled: $X\sim Bin(n, p).$

Given $n=6, p=5/15=1/3, q=1-p=2/3.$

$$P(X>1)=1-P(X=0-P(X=1)$$

$$=1-\dbinom{6}{0}(\dfrac{1}{3})^0(\dfrac{2}{3})^{6-0}-\dbinom{6}{1}(\dfrac{1}{3})^1(\dfrac{2}{3})^{6-1}$$

$$\approx0.65$$

The probability of the batch being sent back for refill is 0.65.