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Discrete Probability Distributions

1. A researcher calculates the expected value for the number of girls in five births. He gets a result of 2.5. He then rounds the result to 3, saying that it is not possible to get 2.5 girls when five babies are born. Is this reasoning correct?
2. A researcher reports that when groups of four children are randomly selected from a population of couples meeting certain criteria, the probability distribution for the number of girl is as given in the accompanying table. Determine whether a probability distribution is given. In those cases where a probability distribution is not described, identify the requirements that are not satisfied. In those cases where a probability distribution is described, find its mean and standard deviation.

X P(x)
0 0.502
1 0.365
2 0.098
3 0.011
4 0.001

3. In a study of brand recognition of Sony, groups of four consumers are interviewed. If x is the number of people in the group who recognize the Sony brand name, then x can be 0,1,2,3, or 4, and the corresponding probabilities are 0.0016, 0.0250, 0.1432, 0.3892 and 0.4096. Is it unusual to randomly select four consumers and find that none of them recognize the brand name Sony?
4. Assume that 12 jurors are randomly selected from a population in which 80% of the people are Mexican Americans. Find the indicated probabilities.

X (Mexican Americans) P(x)
0 0+
1 0+
2 0+
3 0+
4 0.001
5 0.003
6 0.016
7 0.053
8 0.133
9 0.236
10 0.283
11 0.206
12 0.069

a. Find the probability of exactly 5 Mexican-Americans among 12 jurors.
b. Find the probability of 6 or fewer Mexican Americans among 12 jurors.
c. Which probability is relevant for determining whether 6 jurors among 12 is unusually low: the result from part (a) or (b)?
d. Does 6 Mexican Americans among 12 jurors suggest that the selection process discriminates against Mexican Americans? Why or why not?

5. Assume that 12 jurors are randomly selected from a population in which 80% of the people are Mexican Americans. Refer to the table in problem 4.
a. Using the probability values in problem 4, find the probability value that should be used for determining whether the result of 11 Mexican Americans among 12 jurors is unusually high.
b. Does the selection of 11 Mexican American jurors suggest that the selection process favors Mexican American? Why or Why not?

6. For the following determine whether the given procedure results in a binomial distribution. For those that are not binomial, identify at least one requirement that is not satisfied.
a. Treating 50 smokers with Nicorette and recording whether there is a yes response when they are asked if they experience any mouth or throat soreness.

7. Assume that a procedure yields a binomial distribution with a trial repeated with a trial n times. Using a binomial probabilities table, find the probability of x successes given the probability p of success on a given trial.
a. n = 4, x = 3, p = 0.30
b. n = 15, x = 12, p = 0.90

8. Assume that a procedure yields a binomial distribution with a trial repeated n times. Use the binomial probability formula to find the probability of x successes given the probability p of success on a single trial.
a. n = 6, x = 4, p = 0.75

9. Referring to the table below, the probabilities were obtained by entering the values of n = 6 and p = 0.167. In a clinical test of the drug Lipitor, 16.7% of the subjects treated with 10 mg of atorvastatin experienced headaches. In each case, assume that 6 subjects are randomly selected and treated with 10 mg of atorvastatin, then find the indicated probability.

X P (X=x)
0.00 0.3341
1.00 0.4019
2.00 0.2014
3.00 0.0538
4.00 0.0081
5.00 0.0006
6.00 0.0000

a. Find the probability that most two subjects experience headaches. Is it unusual to have at most two of six subjects experience headaches?

10. Mars claims that 20% of its M&M plain candies are orange, and a sample of 100 such candies is randomly selected. Find:
a. Population mean:
b. Population standard deviation
c. Minimum
d. Maximum

11. In a study of 420,095 cell phone users in Denmark, it as found that 135 developed cancer of the brain or nervous system. If we assume that such cancer is not affected by cell phones, the probability of a person having a cancer is 0.000340.
a. Find the population mean:
b. Find population standard deviation:
c. Based on the results from part a and b, is it unusual to find that among 420,095 people, there are 135 cases of cancer of the brain or nervous system? Why or why not?
d. What do these results suggest about the publicized concern that cell phones are a health danger because they increase the risk of cancer of the brain or nervous system?

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