Use the probabilities of Exercise 4.56 to show that these equalities are true: a. P(A © B)

Use the probabilities of Exercise 4.56 to show that these equalities are true:
a. P(A ˆ© B) = P(A)P(B | A)
b. P(A ˆ© B) = P(B)P(A | B)
c. P(A ˆª B) = P(A) + P(B) - P(A ˆ© B)
Exercise 4.56
A study of drug offenders who have been treated for drug abuse suggests that the likelihood of conviction within a 2-year period after treatment may depend on the offender's education. The proportions of the total number of cases that fall into four education/conviction categories are shown in the table below.

Suppose a single offender is selected from the treatment program. Here are the events of interest:
A: The offender has 10 or more years of education
B: The offender is convicted within 2 years after completion of treatment

Transcribed Image Text:

Status Within 2 Years After Treatment Not Education 10 Years or More 9 Years or Less Totals Convicted 10 .27 .37 Convicted .30 .33 .63 Totals 40 60 1.00