### Math 225 Course Notes

### Section 3.3: Elementary Properties of Probability

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.

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%.

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:**

- In what proportion of the patients were their ulcers in remission?
- Of those patients who received a high dose,
what proportion had their condition worsen?
- What proportion of all patients took a placebo and either
had their ulcers go into remission or improve?
- What proportion of patients whose condition was maintained
took the placebo?

**Answers:**
- 14/131 = .107
- 8/43 = .186
- (2+8)/131 = .076
- 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,
larget@mathcs.duq.edu