5.5 – 5.6 Exercises: Conditional Probability and Baye’s Formula

1) Empirical Example: Suppose a survey of 1000 drivers in a metropolitan area during a 3-year period was taken. The following results were found.

Age Group 18-25 26-39 40-55 55+
0-1 Accidents 100 150 250 75 575
2-3 Accidents 150 25 125 25 325
3+ accidents 50 25 25 0 100
Totals 300 200 400 100 1000

 Suppose we randomly select a driver from the group. What is the probability that the driver,

a) Is in the 26-39 age group, given they have more than 3 accidents.

b) Had 2-3 accidents, given they are in the 18-25 age group.

c) Had 0-1 accidents, given they are in the 40-55 age group.

d) Is in the 40-55 age group, given they had 0-1 accidents

2)  Suppose we send 30% of our products to company A and 70% of our products to company B. Company A reports that 5% of our products are defective and company B reports that 4% of our products are defective. (Use a tree diagram)

a) Find the probability that a product is sent to company A and it is defective.

b) Find the probability that a product is sent to company A and it is not defective.

 

c) Find the probability that a product is sent to company B and it is defective.

d) Find the probability that a product is sent to company B and it is not defective.

3) One box has 7 red balls and 3 white balls; a second box has 6 red balls and 4 white balls. A pair of dice are tossed. If the sum of the dice are less than five, a ball is selected from the first box, otherwise the ball is selected from the second box. Find the probability of getting a red ball.

4) Bayes Formula: One urn has 4 red balls and 1 white ball; a second urn has 2 red balls and 3 white balls. A single card is randomly selected from a standard deck. If the card is less than 5 (aces count as 1), a ball is drawn out of the first urn; otherwise a ball is drawn out of the second urn. If the drawn ball is red, what is the probability that it came out of the second urn?

5)  Bayes Formula: A small manufacturing company has rated 75% of its employees as satisfactory (S) and 25% as unsatisfactory (S’). Personnel records show that 80% of the satisfactory workers had previous work experience (E) in the job they are now doing, while 15% of the unsatisfactory workers had no work experience (E’) in the job they are now doing. If a person who has had previous work experience is hired, what is the approximate empirical probability that this person will be an unsatisfactory employee?

6)  Bayes Formula: A basketball team is to play two games in a tournament. The probability of winning the first game is .10. If the first game is won, the probability of winning the second game is .15. If the first game is lost, the probability of winning the second game is .25. What is the probability the first game was won if the second game is lost?

7) Bayes Formula: To evaluate a new test for detecting Hansen’s disease, a group of people 5% of which are known to have Hansen’s disease are tested. The test finds Hansen’s disease among 98% of those with the disease and 3% of those who don’t. What is the probability that someone testing positive for Hansen’s disease under this new test actually has it?