A Pair of Dice Are Rolled

Section 8-3 Conditional Probability, Intersection, and Independence

Example 1. A company has rated 75% of its employees as satisfactory and 25% as unsatisfactory. Personnel records indicate that 80% of the satisfactory workers had previous work experience while only 40% of the unsatisfactory workers had any previous experience. a. If a person with previous work experience is hired, what is the probability that this person will be a satisfactory worker?

Satisfactory Previous Rating Experience Prob PSat  Exp 0.60 0.60 Yes (0.8) 0.60 P(Sat | Exp)     0.857 Yes (0.75) P(Exp) 0.60  0.10 0.70 No (0.2) 0.15

Yes (0.4) 0.10 No (0.25) No (0.6) 0.15 b. If a person with no previous work experience is hired, what is the probability that this person will be a satisfactory employee?

Satisfactory Previous Rating Experience Prob P(Sat  Exp') 0.15 0.15 Yes (0.8) 0.60 P(Sat | Exp')     0.5 Yes (0.75) P(Exp') 0.15  0.15 0.30 No (0.2) 0.15

Yes (0.4) 0.10 No (0.25) No (0.6) 0.15 Example 2. A computer store sells three types of microcomputers; brand A, brand B, and brand C. Of the computers they sell, 60% are brand A, 25% are brand B, and 15% are brand C. They have found that 20% of the brand A computer, 15% of the brand B computers, and 5% of the brand C computers are returned for service during the warranty period.

Brand Defective Prob Yes (0.20) 0.1200 A (0.60) No (0.80) 0.4800

Yes (0.15) 0.0375 B (0.25) No (0.85) 0.2125

Yes (0.05 0.0075 C (0.15) No (0.95) 0.1425 a. If a computer is returned for service during the warranty period, what is the probability that it is a brand A computer?

b. If a computer is returned for service during the warranty period, what is the probability that it is a brand B computer?

c. If a computer is returned for service during the warranty period, what is the probability that it is a brand C computer? Example 3. In a random sample of 200 women who suspect that they are pregnant, 100 turn out to be pregnant. A new pregnancy test given to these women indicated pregnancy in 92 of the 100 pregnant women and in 12 of the 100 women who were not pregnant. a. If a woman suspects she is pregnant and this test indicates that she is pregnant, what is the probability that she really is Pregnant Test Result Prob pregnant? Positive (0.92) 0.46 Yes (0.5) P( preg  pos) 0.46 0.46 Negative (0.08) 0.04 P( preg | pos)     0.885 P( pos) 0.46  0.06 0.52 Positive (0.12) 0.06 No (0.5) Negative (0.88) 0.44

b. If the test indicates that she is not pregnant, what is the probability that she really is pregnant?

P( preg  neg) 0.04 0.04 Pregnant Test Result Prob P( preg | neg)     0.083 P(neg) 0.04  0.44 0.48 Positive (0.92) 0.46 Yes (0.5) Negative (0.08) 0.04

Positive (0.12) 0.06 No (0.5) Negative (0.88) 0.44 Example 4. In a given county, records show that of the registered voters, 45% are Democrats, 35% are Republicans, and 20% are independents. In an election, 70% of the Democrats, 40% of the Republicans, and 80% of the independents voted in favor of a parks and recreation bond proposal.

Party Vote Prob Yes (0.7) 0.315 Dem (0.45) No (0.3) 0.135

Yes (0.4) 0.140 Rep (0.35) No (0.6) 0.210

Yes (0.8) 0.160 Ind (0.20) No (0.2) 0.040 a. If a registered voter chosen at random is found to have voted in favor of the bond, what is the probability that the voter is a Republican?

b. If a registered voter chosen at random is found to have voted in favor of the bond, what is the probability that the voter is an independent?

c. If a registered voter chosen at random is found to have voted in favor of the bond, what is the probability that the voter is a Democrat?