An inspector tests 8 items from a production line…

An inspector tests $8$ items from a production line. Each item is defective with the same probability $p$, independently of the others. She notices that in her records, the probability of finding exactly $2$ defective items is the same as the probability of finding exactly $3$. She wants to estimate the rate of defects from this fact, since her records do not show $p$ directly. Find $p$. Find the mean number of defective items in a sample of $8$. Find the probability of exactly $2$ defective items. The mean will help her plan how many spare items to keep, and the probability will be used in her report. Give decimals to four places. Use the ratio $\dfrac{P(X = 3)}{P(X = 2)}$ to find $p$.