Question

A candy factory produces the candy boxes which include 100 parts in each box. According to the quality control section, 95% of candies are qualified. (a)What is the probability that more than 2 candies in a given box will be non qualified? (b) What is the probability that more than 10 candies in a given box will be non qualified?

Answer 1

$X-number\,\,of\,\,nonqualified\,\,parts,X~Bin\left( 100,0.05 \right) \\p=0.05,n=100\\a:\\\frac{X-100p}{\sqrt{100p\left( 1-p \right)}}\sim N\left( 0,1 \right) \\P\left( X>2 \right) =P\left( \frac{X-100p}{\sqrt{100p\left( 1-p \right)}}>\frac{2-100p}{\sqrt{100p\left( 1-p \right)}} \right) \approx \\\approx P\left( Z>\frac{2-100\cdot 0.05}{\sqrt{100\cdot 0.05\cdot 0.95}} \right) =P\left( Z>-1.37649 \right) =\varPhi \left( 1.37649 \right) =\\=0.9157\\b:\\\frac{X-100p}{\sqrt{100p\left( 1-p \right)}}\sim N\left( 0,1 \right) \\P\left( X>10 \right) =P\left( \frac{X-100p}{\sqrt{100p\left( 1-p \right)}}>\frac{10-100p}{\sqrt{100p\left( 1-p \right)}} \right) \approx \\\approx P\left( Z>\frac{10-100\cdot 0.05}{\sqrt{100\cdot 0.05\cdot 0.95}} \right) =P\left( Z>2.29416 \right) =\varPhi \left( -2.29416 \right) =\\=0.01089$