Question

80. Of women who undergo regular mammograms, two percent have breast cancer. If a woman has breast cancer, there is a 90% chance that her mammogram will come back positive. If she does not have breast cancer there is a 10% chance that her mammogram will come back positive. Given that a woman’s mammogram has come back positive, what is the probability that she has breast cancer?

81. The Triangle is a neighborhood that once housed a chemical plant but has become a residential area. Two percent of the children in the city live in the Triangle, and fourteen percent of these children test positive for the excessive presence of toxic metals in the tissue. For children in the city who do not live in the Triangle, the rate of positive tests is only one percent. If we randomly select a child who lives in the city and she tests positive, what is the probability that she lives in the Triangle?

82. Three percent of Tropicana brand oranges are already rotten when they arrive at the supermarket. In contrast, six percent of Sunkist brand oranges arrive rotten. A local supermarket buys forty percent of its oranges from Tropicana and the rest from Sunkist. Suppose we randomly choose an orange from the supermarket and see that it is rotten. What is the probability that it is a Tropicana?

Answer 1

80) P(positive) = 0.9 * 0.02 + 0.1 * (1 - 0.02) = 0.116

P(breast cancer | positive) = P(positive | breast cancer) * P(breast cancer)/P(positive)

= (0.9 * 0.02)/0.116

= 0.1552

81) P(positive) = 0.14 * 0.02 + 0.01 * (1 - 0.02) = 0.0126

P(lives in the triangle | positive) = P(positive | lives in the triangle) * P(lives in the triangle)/P(positive)

= (0.14 * 0.02)/0.0126 = 0.2222

82) P(rotten) = P(rotten | Tropicana) * P(Tropicana) + P(rotten | Sunkist) * P(Sunkist) = 0.03 * 0.4 + 0.06 * 0.6 = 0.048

P(Tropicana | rotten) = P(rotten | Tropicana) * P(Tropicana)/P(rotten) = (0.03 * 0.4)/0.048 = 0.25