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)=
`(70)/(100)* (69)/(99)* (68)/(98)= (7*69*68)/(10*99*98)=`=0.339

5. If familiar to the Combination formula C(n, r)=
`(n!)/(r!(n-r)!)`:

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