8.2 Warm Up. why not.

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1 Binomial distributions often arise in discrimination cases when the population in question is large. The generic question is If the selection were made at random from the entire population, what is the probability that the number of members of a protected class hired/promoted/laid off would be as small/large as it actually was? This assumes that all members of the qualified population have equal merit, so its just a first step. If the population is large, we can act as if the candidates are chosen independently. In 2004, the National Institute of Health announced that it would give a few new Director Pioneer Awards for research. The awards were highly valued: $500,000 per year for five years for research support. Nine awards were made, all to men. This caused an outcry. There were 1300 nominees for the award, 80% male. Suppose that all nominees are equally qualified. If we choose 9 at random, the number of women among the winners has (to a close approximation) the binomial distribution with n=9 and p=0.2. Call the number of women X. Find P(no award go to women), P(at least one woman), P(no more than one woman), the mean number of women in repeated random drawing, and the standard deviation. Can we use the normal approximation to calculate these probabilities?

2 8.2 Warm Up At an archaeological site that was an ancient swamp, the bones from 20 brontosaur skeletons have been unearthed. The bones do not show any sign of disease or malformation. It is thought that these animals wandered into a deep area of the swamp and became trapped in the swamp bottom. The 20 left femur bones (thigh bones) were located and 4 of these left femurs are to randomly selected without replacement for DNA testing to determine gender. a) Let X be the number out of the 4 selected left femurs that are from males. Based on how these bones were sampled, explain why the probability distribution of X is not binomial. b) Suppose that the group of 20 brontosaurs whose remains were found in the swamp had been made up of 10 males and 10 females. What is the probability that all 4 in the sample to be tested are male? c) The DNA testing revealed that all 4 femurs tested were from males. Based on this result and your answer from part (b), do you think that males and females were equally represented in the group of 20 brontosaurs stuck in the swamp? Explain. d) Is it reasonable to generalize your conclusion from part c) pertaining to the group of 20 brontosaurs to the population of all brontosaurs? Explain why or why not.

3 More discrimination in the workplace There are several thousand workers at a particular factory, of which 30% are Hispanic. We randomly select a sample of 15 employees to serve on a committee to study and recommend changes to the employee benefits program. But only 3 Hispanic employees were selected, and the Hispanic employees have charged that the selection process was rigged to favor non- Hispanics. Is there evidence of this? Specifically, what is the probability that at most 3 Hispanics are chosen for the committee?

4 The Geometric Setting: Used when the goal is to obtain a number of. The random variable X is defined as counting the number of trials needed to obtain that. Possible values of a geometric random variable: 1, 2, 3 (infinite) since it is theoretically possible to proceed indefinitely without ever obtaining a success. Examples:

5 The Geometric Setting: 2PIFS 1. 2: 2. Probability: 3. I: The observations are 4. FS: variable of interest is the number of trials required to obtain the

6 Examples An experiment consists of rolling a single die. The event of interest is rolling a 3; this event is called a success. The random variable X is defined as X = the number of trials until a 3 occurs. Is this a geometric setting? Suppose you repeatedly draw cards without replacement from a deck of 52 cards until you draw an ace. Is this a geometric setting?

7 Probability of success: Probability of failure: Rule for Calculating Geometric Probabilities: Ex. 8.17: An experiment consists of rolling a single die. The event of interest is rolling a 3; this event is called a success. The random variable X is defined as X = the number of trials until a 3 occurs. P(X=1) = P(X=2) = P (X=3) =

8 The Mean (Expected Value) of a Geometric Random Variable: The Variance of a Geometric Random Variable: Example: Glenn likes the game at the fair where you toss a coin into a saucer. You win if the coin comes to rest in the saucer w/o sliding off. Glenn has played this a lot and has determined that he wins 1 out of every 12 times he plays. He believes his chances of winning are the same for each toss. He has no reason to think the tosses are not independent. Let X be the # of tosses until a win. 1) Find the probability of success on any given trial 2) Find the expected number of successes (mean) 3) Find the standard deviation

9 P (X>n): Roll a die until a 3 is observed. Find the probability that it takes more than 6 rolls to observe a 3. Let Y be the number of Glenn s coin tosses until a coin stays in the saucer. The expected number is 12. Find the probability that it takes more than 12 tosses to win a stuffed animal.

10 Calculator for Geometric Distributions: Distr: (p, n) Distr: (p, n)

11 For the offices in a large office building, there are 100 different lock-and-key combinations. You start testing locks to see if the key will fit. The number of locks X you must test to find one that the key fits has a geometric distribution with p = 1/100 = (The necessary assumption here is that each office is equally likely to have any of the 100 combos; this permits us to say that p remains constant at 1/100 on each trial). 1) What is the expected number of offices you will have to visit in order to find an office with a lock that the key fits? 2) What is the probability that you will have to visit at least 200 offices in order to find an office with a lock that the key fits? 3) What is the probability that you will have to visit at most 200 offices?

12 There is a probability of 0.08 that a vaccine will cause a certain side effect. Suppose that a number of patients are inoculated with the vaccine. We are interested in the number of patients vaccinated until the first side effect is observed. 1. Define the random variable of interest. X=? 2. Verify that this describes a geometric setting. 3. Find the probability that exactly 5 patients must be vaccinated in order to observe the first side effect. 4. Construct a probability distribution table for X (up through X = 5). 5. How many patients would you expect to have to vaccinate in order to observe the first side effect? 6. What is the probability that the number of patients vaccinated until the first side effect is observed is at most 5?

13 Exploring Geometric Distributions with the TI83 Page 547 Technology Toolbox. Note: The probability distribution histogram is strongly skewed to the right. The height of each bar after the 1 st is the height of the previous bar times the probability of failure (1-p). Since you are * each consecutive height by a number <1, each new bar will always be shorter than the previous. Therefore the histogram will ALWAYS be right-skewed.