SOLUTION: A study of drug offenders who have been treated for drug abuse suggests that the chance of conviction within a 2-year period after treatment may depend on the offender's education.

Click here to see ALL problems on Probability-and-statistics

Question 1206246: A study of drug offenders who have been treated for drug abuse suggests that the chance 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 following table.
Status Within 2
Years After Treatment
Education Convicted Not
Convicted Totals
10 Years or More 0.12 0.18 0.30
9 Years or Less 0.26 0.44 0.70
Totals 0.38 0.62 1.00
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
Find the appropriate probabilities for these events. (Round your answers to three decimal places.)
(a) A
(b) B
(c) A ∩ B
(d) A ∪ B
(e) AC
(f) A given that B has occurred
(g) B given that A has occurred
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!

Status Within 2 Years After Treatment
Education Convicted Not Convicted Totals
10 Years or More 0.12 0.18 0.3
9 Years or Less 0.26 0.44 0.7
Totals 0.38 0.62 1

A = 10 or more years of education
B = convicted

Answers:
P(A) = 0.3
P(B) = 0.38
P(A ∩ B) = 0.12
P(A U B) = 0.56
P(A^C) = 0.7
P(A given B) = 0.316 (approximate)
P(B given A) = 0.4

Explanation:
Much of the answers are pulled directly from the table without any need to do a computation.
If a computation is performed, then I used the formulas in the next paragraph below.
Almost all of the answers are exact without any rounding done to them. The only exception is that P(A given B) is approximate.

Formulas used:
P(A U B) = P(A) + P(B) - P(A ∩ B) .... Inclusion-Exclusion Principle
P(A^C) = 1 - P(A) .... Probability complement
P(A given B) = P(A ∩ B)/P(B) .... Conditional probability formula
P(B given A) = P(A ∩ B)/P(A)