Question

The pharmacy received 1000 bottles of mineral water. The probability that the bottle

will be broken during transportation is 0.003. Determine the likelihood that the pharmacy will

receive broken bottles: a) Two b) Less than two c) More than two d) At least one

Answer 1

The normal approximation to the binomial distribution:

$\mu=np=1000*0.003=3.$

$\sigma=\sqrt{np(1-p)}=\sqrt{1000*0.003*(1-0.003)}=1.73.$

a) $P(1.5<X<2.5)=P(\frac{1.5-3}{1.73}<Z<\frac{2.5-3}{1.73})=P(-0.87<Z<-0.29)=$

$=P(Z<-0.29)-P(Z<-0.87)=P(Z<0.87)-P(Z<0.29)=0.1937.$

b) $P(X<1.5)=P(Z<\frac{1.5-3}{1.73})=P(Z<-0.29)=0.3859.$

c) $P(X>2.5)=P(Z>\frac{2.5-3}{1.73})=P(Z>-0.87)=0.8078.$

d) $P(X>0.5)=P(Z>\frac{0.5-3}{1.73})=P(Z>-1.45)=0.9265.$