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- Solve basic probability problems.

- Understand probability problems based on enumeration.
- Understand conditional probability.
- Examine contingency tables of categorical variables to informally examine the dependence between the variables.

- According to the National Center for Health Statistics,
the age of the mother for births in the United States in 1992
are distributed as follows.
Age Proportion <15 0.003 15-19 0.124 20-24 0.263 25-29 0.290 30-34 0.220 35-39 0.085 40-44 0.014 45-49 0.001 Total 1.000 A random birth is chosen from all U.S. births in 1992. Find:

- The probability the mother is 24 years old or younger.
- The probability the mother is at least 40 years old.
- The probability the mother is younger than 20 years old given she is younger than 24 years old.
- The probability the mother is younger than 20 years old given she is at least 40 years old.

- In one breed of dogs,
solid coats (S) are dominant over spotted coats (s)
and black color (B) is dominant over tan color (b).
The two genes are inherited independently.
Consider a cross of a black spotted dog with genotype Bbss
with a black solid dog with genotype BbSs that produces a single offspring.
- What is the probability the offspring has a black coat?
- What is the probability the offspring is spotted?
- What is the probability the offspring has a black coat and is spotted?

- Consider the same genes as the previous problem.
Based on pedegree information,
a particular black spotted dog with unknown genotype
is thought to have genotype BBss with probability 1/3
and genotype Bbss with probability 2/3.
The dog is crossed with a tan solid dog with genotype bbSs
and produces a single offspring.
- What is the probability the offspring is black?
- Given the offspring is black, what is the probability that the unknown genotype of the black parent dog is BBss?
- Given the offspring is spotted, what is the probability that the unknown genotype of the black parent dog is BBss?

- The following data are taken from a study investigating
the use of radionuclide ventriculography as a diagnostic test
for detecting coronary artery disease.
Test Disease

PresentDisease

AbsentTotal **Positive**302 80 382 **Negative**179 372 551 **Total**481 452 933 - What are the sensitivity and specificity of the test in this study?
- If a population has a ten percent prevalence of coronary artery disease, what is the probability an individual with a positive test has the disease?
- If a population has a ten percent prevalence of coronary artery disease, what is the probability an individual with a negative test does not have the disease?

- A study finds that a mammogram as a screening test for breast cancer
has a sensitivity of 0.85 and a specificity of 0.80.
(a) What is the false positive rate (probability of a positive result given absence of disease)?

(b) What is the false negative rate (probability of a negative result given presence of disease)?

(c) If the probability of breast cancer is 0.0025, what is the probability of breast cancer given a positive test result?

- A 1992 article found this data
on the socioeconomic status and presence of symptoms
of respiratory illness among infants in North Carolina.
Socioeconomic

StatusNumber of

ChildrenNumber with

SymptomsLow 79 31 Middle 122 29 High 192 27 - Rewrite the table as a crosstabulation of the variables `Socioeconomic status' and `Presence of Symptoms'.
- If an infant is randomly chosen from the study, what is the probability of symptoms being present?
- For each level in the socioeconomic status, find the conditional probability of the presence of symptoms for a randomly selected infant from that group.

- In a group of ten individuals with a medical condition,
three would show improvement when given a placebo
and seven would not.
Five of the ten are randomly sampled and given the placebo.
(a) What is the probability that none of the five sampled individuals shows improvement?

(b) What is the probability that two or more of the sampled individuals show improvement?

- The annual incidence rate of blindness among insulin-dependent diabetic men aged 30-39 is 0.67%. Use the Poisson distribution to calculate the probability that there would be two or fewer cases of blindness in a sample of 250 men from this population.

Bret Larget, larget@mathcs.duq.edu