Question

A robot randomly removes about 1,000 parts each hour from a large bin. For the first hour, the number of parts in the bin is large enough that the removal process is equivalent to sampling with replacement. The parts supplier just made you aware of a production problem; it’s likely that 15% of the parts are unacceptable.

What is the minimum number of parts that would be selected so that the probability is at least 0.50 that at least one part is unacceptable?

Answer 1

Here there are 15% of the parts that are unacceptable.

so here we have to find number of minimum parts to be selected for verification that the probability is at least 0.50 that at least one part is unacceptable.

so here if n is that number so the distribution is binomial where p = 0.15 and if x is the unacceptable parts out of n

so here as per question

Pr(x $\geq$ 1) $\geq$ 0.50

Pr(x $\geq$ 1) =  1 - Pr(x = 0)

Pr(x = 0)  $\leq$ 0.5

so here

Pr(x = 0) = ⁿC₀ (0.15)⁰ (0.85)ⁿ

so here

0.85ⁿ$\leq$ 0.5

n ln (0.85) $\leq$ ln (0.5)

n $\geq$ 4.265

so here atleast 5 parts to be selected to that the probability is at least 0.50 that at least one part is unacceptable.