In this homework you will have to determine if it is exponential or Poisson or binomial.

Remember that for exponential, we are looking at the probability of a time up to some occurrence, or between occurrences, or time to next occurrence of something.

Poisson is the number of occurrences in a FIXED time period.

Both of these involve time being mentioned, so you can’t just glom onto the ‘keyword’ time.

I will either give you a **, which is the average number of counts in a time interval, or µ, which is the average time between occurrences (or to the first occurrence. (remember the diagrams I drew in class). You can use dimensional analysis in the problem to see which is which. The t in the formulas will NEVER be inverted, so it’s either the or the µ that needs to be inverted to fit the units for dimensional analysis.**

**Binomial involves a ‘success’ and a probability of that success, denoted p. It involves the chance of x out of a total of n. n is fixed and finite for binomial. N is ∞ (infinity) for Poisson. **

**In the problems below, if you have an exponential, please draw the relevant diagram with the proper area shaded in.**

**If it is a Poisson or binomial problem with ‘at least’, ‘at most’, etc., sketch the table to ensure you are including the right probabilities in your sum.**

1. In a large department store, the customer service office receives an average of 6 complaints per hour about quality of service.

a. Is the 6 complaints an hour an average time between occurrences, average occurrences per time, or probability of ‘success’?

What Is the probability that in any given hour, exactly 6 complaints will be received?

b. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

c. Find the probability, drawing a diagram for the exponential if that is what you decided on in b.

What is the probability that more than 20 minutes will elapse between successive complaints?

d. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

e. Find the probability, drawing a diagram for the exponential if that is what you decided on in d

What is the probability that less than 5 minutes will elapse between successive complaints?

f. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

g. Find the probability, drawing a diagram for the exponential if that is what you decided on in f.

The store manager observes the complaints office during a 30 minute period, during which NO complaints were received. She concludes that a talk she gave to her staff on the theme “The customer is always right” has obviously had a beneficial effect. Suppose that in fact the talk had NO effect. What is the probability of the manager’s observing a period of 30 minutes or longer with no complaints?

h. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

i. Find the probability, drawing a diagram for the exponential if that is what you decided on in h.

What is the probability of at least 1 complaint in a 30 minute period?

j. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

k. Find the probability, drawing a diagram for the exponential if that is what you decided on in f.

2. A local radio station believes that 70% of its listeners are younger than 25 years of age. 20 listeners are selected at random . What is the probability that more than half of them are younger than 25 years of age, if the radio station’s belief is correct?

a. Is the 70% of listeners younger than 25 years of age an average time between occurrences, average occurrences per time, or probability of ‘success’?

b. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

c. Find the probability, drawing a diagram for the exponential if that is what you decided on in b.

What is the probability that exactly 14 of them are younger than 25 years of age?

d. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

e. Find the probability, drawing a diagram for the exponential if that is what you decided on in d.

f. How many of the 20 listeners sampled would you expect to be younger than 25 years of age?

3. A professor sees students during regular office hours. The average time spent helping each student is 10 minutes.

What is the probability that a professor spends less than 20 minutes with a student?

a. Is the 10 minute average time spent helping each student an average time between occurrences, average occurrences per time, or probability of ‘success’?

b. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

c. Find the probability, drawing a diagram for the exponential if that is what you decided on in b.

What is the chance that the professor helps more than 6 students during office hours?

d. Should you use a binomial, exponential, or Poisson formula to answer this question? How did you decide on the one you chose?

e. Find the probability, drawing a diagram for the exponential if that is what you decided on in d.