Probability problems

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For the following problems, 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 distribution.

1.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 girls is as given in the accompanying table.

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

2. For a group of four men, the probability distribution for the number x who live through the next year is as given in the accompanying table.

x P(x)
0 0.0000
1 0.0001
2 0.0006
3 0.0387
4 0.9606

3.Assume that in a test of gender-selection technique, a clinical trial results in 12 girls in 14 births. Refer to table 4-1 and find the indicated probabilities.

A. Find the probability of exactly 12 girls in 14 births.
B.Find the probability of 12 or more girls in 14 births
C.Which probability is relevant for determining whether 12 girls in 14 births is unusually high: the result from part (a) or part (b)?
D.Does 12 girls in 14 births suggest that the gender-selection technique is effective? Why or why not?

Table 4-1
Probability Distribution
X(girls) P(x)
0 0.000
1 0.001
2 0.006
3 0.022
4 0.061
5 0.122
6 0.183
7 0.209
8 0.183
9 0.122
10 0.061
11 0.022
12 0.006
13 0.001
14 0.000

4. 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.

Treating 50 smokers with Nicorette and asking them if they experience any mouth or throat soreness.

Determining whether each of 3000 heart pacemakers is acceptable or defective.

5. Assume that a procedure yields a binomial distribution with a trial repeated n times. Use Table A-1 to find the probability of x successes given the probability p of success on a given trial.
N= 11, x=7, p=0.05
Use the Poison distribution to find the indicated probabilities.

6. Currently, 11 babies are born in the village of Westport (population 760) each year (based on data from the U.S. National Center for Health Statistics).
a. Find the mean number of births per day
b. Find the probability that on a given day, there are no births.
c. Find the probability that on a given day, there is at least one birth
d. Based on the preceding results, should medical birthing personnel be on permanent standby, or should they be called in as needed? Does this mean that Westport mothers might not get the immediate medical attention that they would be likely to get in a more populated area?

7. Assume that a procedure yields a binomial distribution with n trials and the probability of success for one trial is p. Use the given values of n and p to find the mean u and standard deviation. Also, use the range rule of thumb to find the minimum usual value u-2 standard deviation and the maximum usual value u + 2 standard deviation. (Use Table A-1)
N=767, p=1/6

Table A-1
X P(x)
0 0.410
1 0.410
2 0.154
3 0.026
4 0.002
8. In this care, assume that 6 subjects are randomly selected and treated with 10 mg of atorvastatin, then find the indicated probability. (Use table A-1)
Find the probability that at least one subject experiences headaches. Is it unusual to not have at least one of six subjects experience headaches?