Basic Concepts of Probability
A probability is a number that reflects the chance or likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%. A probability of 0 indicates that there is no chance that a particular event will occur, whereas a probability of 1 indicates that an event is certain to occur. A probability of 0.45 (45%) indicates that there are 45 chances out of 100 of the event occurring.
The concept of probability can be illustrated in the context of a study of obesity in children 5-10 years of age who are seeking medical care at a particular pediatric practice. The population (sampling frame) includes all children who were seen in the practice in the past 12 months and is summarized below.
|
|
Age (years) |
|
|||||
|
|
5 |
6 |
7 |
8 |
9 |
10 |
Total |
|
Boys |
432 |
379 |
501 |
410 |
420 |
418 |
2,560 |
|
Girls |
408 |
513 |
412 |
436 |
461 |
500 |
2,730 |
|
Totals |
840 |
892 |
913 |
846 |
881 |
918 |
5,290 |
Unconditional Probability
If we select a child at random (by simple random sampling), then each child has the same probability (equal chance) of being selected, and the probability is 1/N, where N=the population size. Thus, the probability that any child is selected is 1/5,290 = 0.0002. In most sampling situations we are generally not concerned with sampling a specific individual but instead we concern ourselves with the probability of sampling certain types of individuals. For example, what is the probability of selecting a boy or a child 7 years of age? The following formula can be used to compute probabilities of selecting individuals with specific attributes or characteristics.
P(characteristic) = # persons with characteristic / N
Try to figure these out before looking at the answers:
- What is the probability of selecting a boy? Answer
- What is the probability of selecting a 7 year-old? Answer
- What is the probability of selecting a boy who is 10 years of age? Answer
- What is the probability of selecting a child (boy or girl) who is at least 8 years of age? Answer
Conditional Probability
Each of the probabilities computed in the previous section (e.g., P(boy), P(7 years of age)) is an unconditional probability, because the denominator for each is the total population size (N=5,290) reflecting the fact that everyone in the entire population is eligible to be selected. However, sometimes it is of interest to focus on a particular subset of the population (e.g., a sub-population). For example, suppose we are interested just in the girls and ask the question, what is the probability of selecting a 9 year old from the sub-population of girls? There is a total of NG=2,730 girls (here NG refers to the population of girls), and the probability of selecting a 9 year old from the sub-population of girls is written as follows:
P(9 year old | girls) = # persons with characteristic / N
where | girls indicates that we are conditioning the question to a specific subgroup, i.e., the subgroup specified to the right of the vertical line.
The conditional probability is computed using the same approach we used to compute unconditional probabilities. In this case:
P(9 year old | girls) = 461/2,730 = 0.169.
This also means that 16.9% of the girls are 9 years of age. Note that this is not the same as the probability of selecting a 9-year old girl from the overall population, which is P(girl who is 9 years of age) = 461/5,290 = 0.087.
What is the probability of selecting a boy from among the 6 year olds?