s enter a tax preparer\'s office at an average rate of 11 per hour. They enter r

Question

s enter a tax preparer's office at an average rate of 11 per hour. They enter randomly and independently of one another Express your answers to four decimal places. Express the probabilities as decimal fractions -not as percentages. a. What is the probability that exactly 4 clients will enter the office in the next half hour? b. What is the probability that at least one client enters the office in the next half hour? C. What is the mean of the number of clients who will enter the office in the next half hour? d, what is the variance of the number of clients who will enter the office in the next half hour? Enter your answer in each of the answer

Explanation / Answer

Solution:

We are given = 11 per hour

Here, we have to use Poisson distribution.

P(X=x) = xe- /x!

Part a

We have to find P(X=4)

We have = 11 for one hour

For half hour, = 11/2 = 5.5

P(X=4) = 5.5^4*exp(-5.5)/4!

P(X=4) = 0.155818804

Required probability = 0.1558

Part b

Here, we have to find P(X1)

P(X1) = 1 – P(X<1) = 1 – P(X=0)

P(X=0) = 5.5^0*exp(-5.5)/0!

P(X=0) = 0.004087

P(X1) = 1 – P(X=0)

P(X1) = 1 – 0.004087

P(X1) = 0.995913

Required probability = 0.9959

Part c

Mean for Poisson distribution =

So, mean of the number of clients who will enter the office in the next half hour is 5.5.

Part d

Variance for Poisson distribution =

So, variance of the number of clients who will enter the office in the next half hour is 5.5.

(Note: Mean and variance for Poisson distribution are same.)

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