1. Geometric Distribution
A probability distribution to determine the probability that success will occur on the nth trial of a binomial experiement

2. Geometric Distribution
Repeated binomial trials
Continue until first success
Find probability that first success comes on nth trial
Probability of success on each trial = p

3. Geometric Probability

4. A sharpshooter normally hits the target 70% of the time.
Find the probability that her first hit is on the second shot.
Find the mean and the standard deviation of this geometric distribution.

5. A sharpshooter normally hits the target 70% of the time.
Find the probability that her first hit is on the second shot.
P(2)=p(1-p) n-1 = .7(.3)2-1 = 0.21
Find the mean
 = 1/p = 1/.7 1.43
Find the standard deviation

6. Poisson Distribution
A probability distribution where the number of trials gets larger and larger while the probability of success gets smaller and smaller

7. Poisson Distribution Two outcomes : success and failure
Outcomes must be independent
Compute probability of r occurrences in a given time, space, volume or other interval
 (Greek letter lambda) represents mean number of successes over time, space, area

8. Poisson Distribution

9. The mean number of people arriving per hour at a shopping center is 18.
Find the probability that the number of customers arriving in an hour is 20.
r = 20
 = 18 Find P(20)
e =

10. The mean number of people arriving per hour at a shopping center is 18.

11. Poisson Probability Distribution Table
Table 4 in Appendix II provides the probability of a specified value of r for selected values of .

12. Using the Poisson Table
 = 18, find P(20):

13. Poisson Approximation to the Binomial Distribution
The Poisson distribution can be used as a probability distribution for “rare” events.

14. “Rare” Event
The number of trials (n) is large and the probability of success (p) is small.

15. If n  100 and np < 10, then
The distribution of r (the number of successes) has a binomial distribution which is approximated by a Poisson distribution .
The mean  = np.

16. Use the Poisson distribution to approximate the binomial distribution:
Find the probability of at most 3 successes.

17. Using the Poisson to approximate the binomial distribution for n = 240 and p = 0.02
Note that n  100 and np = 4.8 < 10, so the Poisson distribution can be used to approximate the binomial distribution.
Find the probability of at most 3 successes:
Since  = np = 4.8, we use Table 4 to find
P( r  3) = = .2942