Practice problems for these concepts can be found at:

- Probability Solved Problems for Beginning Statistics
- Probability Supplementary Problems for Beginning Statistics

Table 4.2 classifies the 500 members of a police department according to their minority status as well as their promotional status during the past year. One hundred of the individuals were classified as being a minority and seventy were promoted during the past year. The probability that a randomly selected individual from the police department is a minority is = .20 and the probability that a randomly selected person was promoted during the past year is = .14. Table 4.3 is obtained by dividing each entry in Table 4.2 by 500.

The four probabilities in the center of Table 4.3, .70, .16, .10, and .04, are called *joint probabilities*. The four probabilities in the margin of the table, .80, .20, .86, and .14, are called *marginal probabilities*.

The joint probabilities concerning the selected police officer may be described as follows:

.70 = the probability that the selected officer is not a minority and was not promoted

.16 = the probability that the selected officer is a minority and was not promoted

.10 = the probability that the selected officer is not a minority and was promoted

.04 = the probability that the selected officer is a minority and was promoted

The marginal probabilities concerning the selected police officer may be described as follows:

.80 = the probability that the selected officer is not a minority

.20 = the probability that the selected officer is a minority

.86 = the probability that the selected officer was not promoted during the last year

.14 = the probability that the selected officer was promoted during the last year

In addition to the joint and marginal probabilities discussed above, another important concept is that of a *conditional probability*. If it is known that the selected police officer is a minority, then the conditional probability of promotion during the past year is = .20, since 100 of the police officers in Table 4.2 were classified as minority and 20 of those were promoted. This same probability may be obtained from Table 4.3 by using the ratio = .20.

The formula for the conditional probability of the occurrence of event A given that event B is known to have occurred for some experiment is represented by P(A | B) and is the ratio of the joint probability of A and B divided by the probability of B. The following formula is used to compute a conditional probability.

- P(A | B) = (

*4.6*)

The following example summarizes the above discussion and the newly introduced notation.

**EXAMPLE 4.15** For the experiment of selecting one police officer at random from those described in Table 4.2, define event A to be the event that the individual was promoted last year and define event B to be the event that the individual is a minority. The joint probability of A and B is expressed as P(A and B) = .04. The marginal probabilities of A and B are expressed as P(A) = .14 and P(B) = .20. The conditional probability of A given B is P(A | B) = = = .20.

Practice problems for these concepts can be found at:

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