### 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.

Questions:

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?