Question

Four friends, Janice, Barbara, Kathy and Roberta, decided to carpool together to get to school. Each day the driver would be chosen by randomly selecting one of the four names. They carpool to school for 92 days. Use the normal approximation to the binomial to calculate the following probabilities. Round the standard deviation to four decimal places. (Round your answers to four decimal places.)

(a) Find the probability that Janice is the driver at most 17 days.


(b) Find the probability that Roberta is the driver more than 14 days.


(c) Find the probability that Barbara drives exactly 23 of those 92 days.

Answer 1

n=	92	p=	0.2500
here mean of distribution=μ=np=		23
and standard deviation σ=sqrt(np(1-p))=		4.1533
for normal distribution z score =(X-μ)/σx
therefore from normal approximation of binomial distribution and continuity correction:

a)

probability that Janice is the driver at most 17 days :

probability =

P(X<17.5)

=

P(Z<-1.32)=

0.0934

b)

probability that Roberta is the driver more than 14 days :

probability =

P(X>14.5)

=

P(Z>-2.05)=

1-P(Z<-2.05)=

1-0.0204=

0.9798

c)

probability that Barbara drives exactly 23 of those 92 days :

probability =

P(22.5<X<23.5)

=

P(-0.12<Z<0.12)=

0.5478-0.4522=

0.0956