for the n=242 Volunteers in a University Study of Rapid Eye Movement (REM)
| Gender | Age | |||
|---|---|---|---|---|
| <18 | 18–20 | 21–25 | >25 | |
| Female | 3 | 58 | 42 | 25 |
| Male | 1 | 61 | 43 | 9 |
1 a female volunteer is selected,
2 a male volunteer younger than 21 is selected,
3 a male volunteer or a volunteer older than 25 is selected.
Solutions Let F be the event a female volunteer is selected, M the event a male volunteer is selected, 21 the event a volunteer younger than 21 is selected, and 25 the event that a volunteer older than 25 is selected. Since a volunteer will be selected at random, each volunteer is equally likely to be selected, and thus,
1 the probability that a female volunteer is selected is
2 the probability that a male volunteer younger than 21 is selected is
3 the probability that a male volunteer or a volunteer older than 25 is selected is
Example 2.21
Use table of percentages for blood type and Rh factor given in Table 2.9 to determine the probability that a randomly selected individual
Table 2.9 The Percentages of Each Blood Type and Rh Factor
| Blood Type | Rh Factor | |
|---|---|---|
| + | − | |
| O | 38% | 7% |
| A | 34% | 6% |
| B | 9% | 2% |
| AB | 3% | 1% |
1 has Rh positive blood,
2 has type A or type B blood,
3 has type A blood or Rh positive blood.
Solutions Based on the percentages given in Table 2.9
1 Rh positive blood is O+, A+, B+, and AB+, and thus, the probability that a randomly selected individual has Rh positive blood is
2 type A or type B blood is A+, A−, B+, B−. Thus,
3 the probability an individual has type A blood or Rh positive blood is
2.3.2 Conditional Probability
In many biomedical studies, the probabilities associated with a qualitative variable will be modeled. A good probability model will take into account all of the factors believed to cause or explain the occurrence of the event. For example, the probability of survival for a melanoma patient depends on many factors including tumor thickness, tumor ulceration, gender, and age. Probabilities that are functions of a particular set of conditions are called conditional probabilities.
Conditional probabilities are simply probabilities based on well-defined subpopulations defined by a particular set of conditions (i.e., explanatory variables). The conditional probability of the event A given that the event B has occurred is denoted by P(A|B) and is defined as
provided that P(B)≠0. Specifying that the event B has occurred places restrictions on the outcomes of the chance experiment that are possible. Thus, the outcomes in the event B become the subpopulation upon which the probability of the event A is based.
Example 2.22
Suppose that in a population of 100 units 35 units are in event A, 48 of the units are in event B, and 22 units are in both events A and B. The unconditional probability of event A is P(A)=35100=0.35. The conditional probability of event A given that event B has occurred is
In this example, knowing that event B has occurred increases the probability that event A will occur from 0.35 to 0.46. That is, event A is more likely to occur if it is known that event B has occurred.
Since conditional probabilities are probabilities, the rules associated with conditional probabilities are similar to the rules of probability. In particular,
1 conditional probabilities are always between 0 and 1. That is, 0≤P(A|B)≤1.
2 P(A does not occur|B)=1−P(A|B).
3 P(A or B|C)=P(A|C)+P(B|C)−P(A and B|C).
Conditional probabilities play an important role in the detection of rare diseases and the development of screening tests for diseases. Two important conditional probabilities used in disease detection are the sensitivity and specificity. The sensitivity is defined to be the conditional probability of a positive test for the subpopulation of individuals having the disease (i.e., P(+|D)), and the specificity is defined to be the conditional probability of a negative test for the subpopulation of individuals who do not have the disease (i.e., P(−|not D)). Thus, the sensitivity of a diagnostic test measures the accuracy of the test for a individual having the disease, and the specificity measures the accuracy of the test for individuals who do not have the disease.
The probability that an individual will test positive in a diagnostic test is a weighted average of the proportion of correct diagnoses (i.e., true positives) and the false diagnoses (i.e., false positives). The formula for computing the probability that an individual tests positive is
A good diagnostic test for detecting a disease must have high sensitivity and specificity