# What is Probability?

With regard to health outcomes probability is the proportion or percentage of events that occur in a group of people, i.e., the number of health outcomes or "events" divided by the total number of possible events.

Probabilities can be expressed as:

• A proportion from 0 to 1, e.g., 0.32 or
• A percentage, e.g., 0.32=32%

Example:

In the Physicians' Health Study on aspirin, male physicians were randomly assigned to take an aspirin or a placebo every other day.

 MI No MI Total Probability Aspirin 126 10,911 11,037 126/11,037 = 0.0114= 1.14% Placebo 213 10,821 11,034 213/11,034 = 0.0193= 1.93%

Test Yourself

Consider the data in the table below from the New York City Cancer Registry

 Cancer Type White Male White Female Black Male Black Female Total Colorectal 1,236 1,251 449 584 3,520 Liver 330 134 149 57 670 Lung 1,449 1,332 537 497 3,815 Thyroid 175 537 29 135 876 Non-Hodgkins 582 523 170 159 1,434 Leukemia 348 285 87 85 805 TOTALS 4,120 4,062 1,421 1,517 11,120

What is the:

1. Probability that a cancer patient is a black male? [P(Black male)]
2. Probability that a cancer patient has thyroid cancer? [P(thyroid cancer)]
3. Probability that a cancer patient is a white female with liver cancer? [P(White female with liver cancer)]
4. Probability that a cancer patient is a black with lung cancer? [P(Black patient with lung cancer)]

Work these out yourself before looking at the answers.

## Conditional Probability

Conditional probability is the probability of an event occurring, given that another condition is true or another event has occurred.

Example: What proportion of patients with colorectal cancer are white?

Another way of stating this question is "Given that a patient has colorectal cancer, what is the probability that the patient is white?

The notation for this is P(white | colorectal cancer), where the "|" signals the condition. In this case, P(white | colorectal cancer) = (1236 + 1251)/3520 = 0.7065 = 71%.

Test Yourself

Using the data in the table above:

1. What proportion of patients with leukemia are white?
2. What proportion of women have lung cancer?

Work these out yourself before looking at the answers.