Probability Practice Problems 1.

Hyperlipidemia in children has been hypothesized to be related to high cholesterol in their parents. The following data were collected on parents and children. CHILD Not Hyperlipidemic Hyperlipidemic

Both Parents Hyperlipidemic 13 45

One Parent Hyperlipidemic 34 42

Neither Parent Hyperlipidemic 83 6

a. What is the probability that one or both parents are hyperlipidemic? b. What is the probability that the child and both parents are hyperlipidemic? c. What is the probability that a child is hyperlipidemic IF neither of his/her parents are hyperlipidemic? d. What is the probability that a child is hyperlipidemic IF both of his/her parents are hyperlipidemic? 2.

A national survey of graduate students is conducted to assess their consumption of coffee. The following table summarizes the data.

Male Female

Do not drink coffee 145 80

Drink Decaffeinated Only 94 121

Drink Caffeinated Coffee 365 430

a. What proportion of students drink decaffeinated coffee only? b. What proportion of coffee drinkers (caffeinated and decaffeinated) are female? c. What proportion of the females do not drink coffee?

3.

Investigators want to assess the accuracy of self-reported smoking status. Participants are asked whether they currently smoke or not. In addition, laboratory tests are performed on hair samples to determine presence or absence of nicotine. The laboratory assessment is considered the gold standard, or truth about nicotine. The data are as follows:

Self-Reported Non-Smoker Self-Reported Smoker

Nicotine Absent 82 12

a. What is the sensitivity of self-reported smoking status? b. What is the specificity of self-reported smoking status?

Nicotine Present 14 52

4.

The following table displays blood pressure status by sex.

Male Female Total

Optimal 22 43 65

Normal 73 132 205

Hypertension 55 65 120

Total 150 240 390

a. What proportion of the participants have optimal blood pressure? b. What proportion of men have optimal blood pressure? c. What proportion of participants with hypertension are male? P(Male | Hypertension) = 55/120 = 0.458. d. Are hypertensive status and male gender independent?

5.

The following table cross classifies women in the study by their body mass index (BMI) at 16 weeks gestation and whether they had preterm delivery.

Preterm Full Term

BMI < 30 320 4700

BMI 30-34.9 80 480

BMI 35+ 120 300

a. What is the probability that a woman delivers preterm? b. What is the probability a women has BMI <30 and delivers preterm? c. What proportion of women with BMI 35+ deliver preterm? d. Are BMI and preterm delivery independent? Justify.

6.

A case-control study is conducted to assess the relationship between heavy alcohol use during the first trimester of pregnancy and miscarriage. Fifty women who suffered miscarriage are enrolled along with 50 who delivered full term. Each participantâ€™s use of alcohol during pregnancy is ascertained. Heavy drinking is defined as 4 or more drinks on one occasion. The data are shown below.

Heavy Alcohol Use No Heavy Alcohol Use

Miscarriage 14 36

Delivered Full Term 4 46

a. Compute the odds of miscarriage in women with heavy alcohol use in pregnancy. b. Compute the odds of miscarriage in women with no heavy alcohol use in pregnancy. c. Compute is the odds ratio for miscarriage as a function of heavy alcohol use.

7.

A randomized trial is conducted to evaluate the efficacy of a new cholesterol lowering medication. The primary outcome is incident coronary artery disease. Participants are free of coronary artery disease at the start of the study and randomized to receive either the new medication or a placebo. Participants are followed for a maximum of 10 years for the development of coronary artery disease. The following data are observed. Number of Participants

Cholesterol Medication Placebo

400 400

Number with Coronary Artery Disease 28 42

a. Compute the relative risk of coronary artery disease for patients receiving the new medication as compared to placebo. b. Compute the odds ratio of coronary artery disease for patients receiving the new medication as compared to placebo.