Practice for exponential distribution and binomial approximation

Exponential

P(x<= a) = 1-e-λa

1. If a salesperson averages 2 sales per week, what are the chances that he will go more than 2 days without a sale?

43% chance that less than 2 days go by between sales, and a 57% chance of more than 2 days between sales

(2/7) = .28 sales per day

2. If a vacuum cleaner lasts an average of 10 years without a breakdown needing repair. What are the odds that it will out-last its 5 year warranty?

39 percent chance of failure in less than 5 years. 61 percent chance that failure occurs after 5 years.

What are the odds that it will out-last its 5 year extended warranty?

there is a 63% chance that it will break in under 10 years, and a 37 percent chance that it will last over 10 years

What are the chances that it will fail during the extended warranty if it has not failed during the standard warranty?

61

37

24

5 10

there is a 61 percent chance of failure after 5, 37 after 10 so 24 between 5 and 10.

3. Expansion teams in the NFL wins 3.7 games per season (16 games), what are the chances that an expansion team will go “winless” in one season?

there is a 97 percent chance that a win will occur in less than one season, and a 2.5% chance that it will take more than one season to get a win.

(this seems better as a binomial, but p not constant)

Binomial Approximation

1. Over the last 5 years the Phillies have won 28 percent of their games. What are the chances that they would win at least 40 of 60 games without improving?

Using binomial appprox:

3*3.48 = 10.44 (three standard deviations)

16.8+/- 10.44 is greater than 0 and less than 60 so we are o.k. with the approximation

16.8 39.5

There is a 100 percent chance that they will win less than 40 games if they are still winning only 28 percent of their games. (no chance they will win that many if they have not improved)

2. Bob drives to school every day 5 days a week for 15 weeks. He finds a spot in the good lot 20 percent of the time. What are the chances that next semester he gets a spot more than 20 times?

Using binomial appprox:

3*3.46 = 10.39 (three standard deviations)

15+/- 10.39 is greater than 0 and less than 75 so we are o.k. with the approximation

15 20.5

There is approximately a 5.6% chance that he will get the good lot more than 20 times.