# Math 225

## Probability

#### Prerequisites

This lab assumes that you already know how to:
1. login, find course Web page, run S-PLUS
2. use the Commands Window to execute commands

#### Technical Objectives

This lab will teach you to:
1. Solve basic probability problems.

#### Conceptual Objectives

In this lab you should learn to:
1. Understand probability problems based on enumeration.
2. Understand conditional probability.
3. Examine contingency tables of categorical variables to informally examine the dependence between the variables.

The lab begins with a number of elementary probability problems to solve by hand.
1. 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.

AgeProportion
<150.003
15-190.124
20-240.263
25-290.290
30-340.220
35-390.085
40-440.014
45-490.001
Total1.000

A random birth is chosen from all U.S. births in 1992. Find:

1. The probability the mother is 24 years old or younger.
2. The probability the mother is at least 40 years old.
3. The probability the mother is younger than 20 years old given she is younger than 24 years old.
4. The probability the mother is younger than 20 years old given she is at least 40 years old.

2. 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.
1. What is the probability the offspring has a black coat?
2. What is the probability the offspring is spotted?
3. What is the probability the offspring has a black coat and is spotted?
3. 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.
1. What is the probability the offspring is black?
2. Given the offspring is black, what is the probability that the unknown genotype of the black parent dog is BBss?
3. Given the offspring is spotted, what is the probability that the unknown genotype of the black parent dog is BBss?
4. The following data are taken from a study investigating the use of radionuclide ventriculography as a diagnostic test for detecting coronary artery disease.

TestDisease
Present
Disease
Absent
Total
Positive30280382
Negative179372551
Total481452933

1. What are the sensitivity and specificity of the test in this study?
2. 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?
3. 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?

#### Homework Assignment

1. 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?

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

Socioeconomic
Status
Number of
Children
Number with
Symptoms
Low7931
Middle12229
High19227

1. Rewrite the table as a crosstabulation of the variables `Socioeconomic status' and `Presence of Symptoms'.
2. If an infant is randomly chosen from the study, what is the probability of symptoms being present?
3. For each level in the socioeconomic status, find the conditional probability of the presence of symptoms for a randomly selected infant from that group.

3. 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?

4. 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.