Question

in a batch of 15 jars f instant coffee, 5 have been under filled. suppose of these jars are selected without replacement. if more than one of these selected jars are under filled, the whole batch is sent back to refill. what is the probability 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.