Question

A manufacturing process has a? 70% yield, meaning that? 70% of the products are acceptable and? 30% are defective. if threethree of the products are randomly? selected, find the probability that all of them are acceptable. round to threethree decimal places as needed.

Answer 1

1. Assume the manufacturer produced 100 products.

2. 70 are acceptable and 30 are defective.

3. the first product has a probability of 70/100 to be selected from the acceptable ones. The second 69/99 (one is removed from the good ones) and the third one 68/98

4. so P(pick 3 consecutive acceptable products)=$\frac{70}{100}* \frac{69}{99}* \frac{68}{98}= \frac{7*69*68}{10*99*98}=$=0.339

5. If familiar to the Combination formula C(n, r)=$\frac{n!}{r!(n-r)!}$:

P(picking 3 acceptable out of 100)=n(picking 3 acceptable)/n(picking 3)=C(70, 3)/C(100/3)=$\frac{70!}{3!67!}/\frac{100!}{3!97!}=\frac{70*69*68*67!}{3!67!}/\frac{100*99*98*97!}{3!97!}= \frac{70*69*68}{3!}/\frac{100*99*98}{3!}= \frac{70*69*68}{100*99*98}$ = 0.339