Warm Up

1. Are the following binomial problems (prove it)? Find the probabilities and decide whether or not the events described are unusual. Find the probability of getting exactly 3 girls when 5 babies are born. Find the probability of getting at most 2 girls when 5 babies are born. Find the probability of getting 3 lefties in a class of 15 given 10% of us are lefties (use the formula). US Air had 20% of all domestic flights and was involved in 4 of 7 consecutive crashes. Find the probability that when 7 airliners crash, 4 are from US Air. Warm Up

2. Section 5-4 Mean, Variance, and Standard Deviation for the Binomial Distribution

3. Key Concept In this section we consider important characteristics of a binomial distribution including center, variation and distribution. That is, we will present methods for finding its mean, variance and standard deviation. As before, the objective is not to simply find those values, but to interpret them and understand them.

4. Meanµ = [x•P(x)] Variance2= [x2•P(x) ] –µ2 Std. Dev = [x2•P(x) ] –µ2 For Any Discrete Probability Distribution: Formulas

5. Meanµ = n•p Variance2 = n•p•q Std. Dev. = n • p • q Binomial Distribution: Formulas Where n = number of fixed trials p = probability of success in one of the n trials q = probability of failure in one of the n trials

6. Maximum usual values = µ + 2 Minimum usual values = µ– 2  Interpretation of Results It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits:

7. Assuming that the jurors from our previous example are randomly selected from a population that is 80% Mexican-American (n=12, p = 0.8), use formulas 5-6 and 5-8 to find the mean and standard deviation for the numbers of Mexican-Americans on juries selected from this population. During a period of 11 years, 870 grand jurors were randomly selected for duty. Find the mean and standard deviation for the numbers of Mexican-Americans. Use the range rule of thumb to find the minimum usual number and the maximum usual number of Mexican-Americans. Based on those numbers, can we conclude that the actual result of 339 Mexican-Americans is unusual? Does this suggest that the selection process discriminated against Mexican-Americans? Example

8. In a test of gender-selection technique, 150 couples each have one baby, and the results consist of 100 girls and 50 boys. Find the mean and standard deviation for the numbers of girls that would occur in groups of 150 births. Is the result of 100 girls unusual? Why or why not? You!

9. Recap In this section we have discussed: • Mean,variance and standard deviation formulas for the any discrete probability distribution. • Mean,variance and standard deviation formulas for the binomial probability distribution. • Interpreting results.

10. Section 5-5 The Poisson Distribution

11. Key Concept The Poisson distribution is important because it is often used for describing the behavior of rare events (with small probabilities). Describes behavior for things such as: radioactive decay, arrivals of people in a line, eagles nesting in a region, patients arriving at an emergency room, and Internet users logging onto a Web site. For example, suppose your local hospital experiences a mean of 2.3 patients arriving at the emergency room between 10 and 11 pm. We can find the probability that for a randomly selected Friday between 10 and 11 pm, exactly 4 patients arrive.

12. The Poisson distribution is a discrete probability distribution that applies to occurrences of some event over a specified interval. The random variablex is the number of occurrences of the event in an interval. The interval can be time, distance, area, volume, or some similar unit. Formula µx• e -µ P(x) = wheree  2.71828 x! Definition

13. The random variable x is the number of occurrences of an event over some interval. The occurrences must be random. The occurrences must be independent of each other. The occurrences must be uniformly distributed over the interval being used. Parameters The mean is µ. • The standard deviation is  = µ . Poisson Distribution Requirements

14. Difference from a Binomial Distribution The Poisson distribution differs from the binomial distribution in these fundamental ways: • The binomial distribution is affected by the sample size n and the probability p, whereas the Poisson distribution is affected only by the mean μ. • In a binomial distribution the possible values of the random variable x are 0, 1, . . . n, but a Poisson distribution has possible x values of 0, 1, . . . , with no upper limit.

15. In analyzing hits by V-1 buzz bombs in World War II, South London was subdivided into 576 regions, each with an area of 0.25 km-squared. A total of 535 bombs hit the combined area of 576 regions. If a region is randomly selected, find the probability that it was hit exactly twice. Based on the probability found in part (a), how many of the 576 regions are expected to be hit exactly twice? Example

16. V-1 Buzz Bomb Hits for 576 Regions in South London

17. Rule of Thumb n 100 np 10 Poisson as Approximation to Binomial The Poisson distribution is sometimes used to approximate the binomial distribution when n is large and p is small.

18. n 100 np 10 Poisson as Approximation to Binomial - μ Value for μ = n • p

19. In Kentucky’s Pick 4 game, you pay $1 to select a sequence of four digits, such as 2283. If you play this game once every day, find the probability of winning exactly once in 365 days. Example

20. A barber finds that on Fridays between 4:00 pm and 5:00 pm, the mean number of arrivals is 6.0. If the arrivals follow a Poisson distribution, find the probability of getting exactly 3 arrivals during that time. Is it unusual to get exactly 3 arrivals? You!

21. Recap In this section we have discussed: • Definition of the Poisson distribution. • Requirements for the Poisson distribution. • Difference between a Poisson distribution and a binomial distribution. • Poisson approximation to the binomial.