Question

A lot of parts contains 500

items, 100 of which are defective. Suppose that 20 items are the number of selected items that are defective.
selected at random. Let X be Express the quantity b. Use the binomial P(X = 5) using factorials. approximation to compute an approximation to P(X = 5).

Answer 1

First, we need to calculate the probability of selecting a defective item, which is given as 100 defective items out of 500 total items, or a probability of 0.2.

Next, we can use the binomial probability formula to calculate P(X = 5) = (20 choose 5) * (0.2)^5 * (0.8)^15 = 0.026.

Step-by-step explanation:

To calculate the probability of selecting exactly 5 defective items out of a total of 20 items, we can use the binomial distribution formula.

First, we need to calculate the probability of selecting a defective item, which is given as 100 defective items out of 500 total items, or a probability of :

100/500

= 0.2.

Next, we can use the binomial probability formula, which is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the total number of items selected, k is the number of defective items selected, and p is the probability of selecting a defective item.

P(X = 5) = (20 choose 5) * (0.2)^5 * (0.8)^(20-5)

Using factorials, (20 choose 5)

= 20! / (5! * (20-5)!)

= 15504

Substituting the values, P(X = 5)

= 15504 * (0.2)^5 * (0.8)^15

= 0.026.