Question

The IRS offers taxpayers the choice of allowing the IRS to compute the amount of their tax refund. During the busy filing season, the number of returns received at the Springfield Service Center that request this service follows a Poisson distribution with a mean of three per day. What is the probability that on a particular day:
a. There are no requests?
b. Exactly three requests appear?
c. Five or more requests take place?
d. There are no requests on two consecutive days?

Answer 1

Let X follows a Poisson distribution with parameter λ = 3.

a)

Calculate the probability that there are no requests.

P(No Request) = P(X = 0)

                          = e-3e0/0!

                         = 0.0498 × 1/1

                         = 0.0498

 

Hence, the required probability is, 0.0498.

 

b)

Calculate the probability that there exactly three requests are appear.

P(Exactly 3 request) = P(X = 3)

                                    = e-3e3/3!

                                   = 0.0498 × 27/6

                                   = 0.2240

 

Hence, the required probability is, 0.2240.

 

c)

Calculate the probability that there are 5 or more requests take a place.

 

P(5 or more requests) = P(X ≥ 5)

                                        = 1 – P(X < 5)

                                       = 1 – {P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)}

                                       = 1 – (e-3e0/0! + e-3e0/1! + e-3e0/2! + e-3e0/3! + e-3e0/4!)

                                       = 1 – (0.0498 + 0.1494 + 0.2240 + 0.2240 + 0.1680)

                                       = 1 – 0.8153

                                       = 0.1847

 

Hence, the required probability is, 0.1847.

 

(d)

The probability that there are no requests on two consecutive days is,

0.0498 ∙ 0.0498 = 0.00248004

 

Hence, required probability is 0.0025.

a) Hence, the required probability is, 0.0498.

b) Hence, the required probability is, 0.2240.