Question

The Post Office has established a record in a major Midwestern city for delivering 90 percent of its local mail the next working day. If you mail eight local letters:

a) What is the probability that all of them will be delivered the next day?

b) What is the average number you expect to be delivered the next day?

c) Calculate the standard deviation of the number delivered when 8 local letters are mailed.

d) What is the probability that the number delivered will be within 2 standard deviations of the mean?

Answer 1

a)

According to the question, the post office delivers 90% of its local mails the next working day.

So the probability that a letter will be delivered the next working day is :

P( a letter will be delivered the next working day ) =90/100 = 0.9

Hence the probability that all 8 letters will be delivered the next working day is :

P( all 8 letters will be delivered the next working day )

= [P( a letters will be delivered the next working day ) ]⁸

=(0.9)⁸ = 0.43046 ~ 0.4304

b)

Here we have to find average number mails expect to be delivered the next day,

We know the formula of average number is,

E(X) = np = 8 * 0.9 = 7.2

average number mails expect to be delivered the next day = 7.2

c)

Here we have to find the  standard deviation of the number delivered when 8 local letters are mailed.

standard deviation , σ = √np(1-p)

here n = 8 , p = 0.9

σ = √8 * 0.90 (1-0.9) = √0.72 = 0.8485 ~ 0.85

standard deviation of the number delivered when 8 local letters are mailed σ = 0.85

d)

X~Bin(8,0.9)

mean=np=8*.9=7.2

variance=np(1-p)=8*0.1*0.9=0.72

standard dev=sqrt(var)=0.8485281

mean-2*sd=7.2-2*0.8485281= 5.502944

mean+2*sd=7.2+2*0.8485281=8.897056

P(5.502944<X<8.897056)=

P(X=6) P(X=7) P(X=8)=8C6*(0.9)^6*(0.1)^2 * 8C7*(0.9)^7(0.1) * 8C8*(0.9)^8 =0.9619

probability that the number delivered will be within 2 standard deviations of the mean, = 0.9619