Math 225 Course Notes

Section 3.3: Elementary Properties of Probability

Key Concepts

Probability measures uncertainty on a scale from 0 to 1 where 0 represents things that have no chance of occuring, and 1 represents things that are certain to occur. We will study probability in the context of random sampling and the binomial and normal distributions. Cross-classified data is a way to tabulate two (or more) categorical variables. We will learn how to determine simple proportions from data in this form.

Basic Probability

Probabilities are similar to proportions. They are always between 0 and 1. A probability represents the likelihood of something occuring. 0 is the probability of events that have no chance of occuring, and 1 is the probability of events that are certain to occur.

If two events cannot both be true, then the probability that one occurs is the sum of the probabilities.

Example: The probability of randomly choosing a person who has missed 0 or 1 days of work due to illness in the past three months is the probability of choosing a person who missed 0 days plus the probability of choosing a person who missed 1 day.

If two events do not depend on each other, the probability that both happen is the product of their probabilities.

Example: Suppose that 40% of all adults in the U.S. have high cholesterol levels. If two adults are randomly chosen, then the probability that both have high cholesterol is .4 times .4 = .16, or 16%.

Cross-classified Data

Here is data from a study on the effects of the drug Mesalamine on patients with mild ulcer problems. The two variables are outcome and treatment . There were a total of 131 individuals in the study.
                      Treatment Group
Outcome         Placebo   Low Dose   High Dose
In remission       2          6           6   |  14
Improved           8         13          15   |  36
Maintained        12         11          14   |  37
Worsened          22         14           8   |  44
Totals            44         44          43   | 131
The outcome is the status of the ulcer after a period of six weeks.


  1. In what proportion of the patients were their ulcers in remission?
  2. Of those patients who received a high dose, what proportion had their condition worsen?
  3. What proportion of all patients took a placebo and either had their ulcers go into remission or improve?
  4. What proportion of patients whose condition was maintained took the placebo?
  1. 14/131 = .107
  2. 8/43 = .186
  3. (2+8)/131 = .076
  4. 12/37 = .324
You should be able to answer similar kinds of questions for similar tables of data.
Last modified: Feb 19, 1996

Bret Larget,