MTH 105 A & C, Spring 2024 Tree Diagrams: Problems

4. (Homework.) In a certain city, 46% of registered voters belong to party I, 30% to party D, and the rest to R. In a recent election, I was represented at the polls with 35% of its membership, compared to 62% for D, and 58% for R.

a. What is the probability that a registered voter actually voted?

b. Given that a registered voter actually voted, what is the conditional probability that they are a member of party I?

5. (Homework.) Three different suppliers, X, Y , and Z, provide produce for a grocery store. Twelve percent of produce from X is superior grade, 8% of produce from Y is superior grade, and 15% of produce from Z is superior grade. The store obtains 20% of its produce from X, 45% from Y , and 35% from Z.

a. If a piece of produce is purchased, what is the probability that it is superior grade?

b. Given that a piece of produce in the store is of superior grade, what is the condi- tional probability that it was supplied by X?

6. (Homework.) The test for a certain disease is 95% accurate for those who actually have the disease and 99% accurate for those who do not. Assume that 3% of the population has the disease.

a. What is the probability that a random member of the population gets a negative test for the disease?

b. Given that a person really has the disease, what is the conditional probability that they get a positive test result?

c. Given that a person gets a positive test result, what is the conditional probability that they really have the disease?

d. Given that a person gets a negative test result, what is the conditional probability that they really do not have the disease?

8. (Homework.) Suppose that we have two urns, A and B. Urn A holds 4 red and 6 green marbles. Urn B holds 5 red and 5 green marbles. An urn is picked at random and then a random marble is selected from the urn. The marble is set aside, and one of the opposite color (from an extras pile) is put back in the urn. Finally, a second marble is drawn from the urn (the same urn that was selected at the start of the process).

a. What is the probability that the first marble that is drawn is green?

b. What is the probability that the second marble that is drawn is red?

c. Given that the second marble that is drawn is red, what is the conditional prob- ability that the urn that was selected at the start was B?

d. Given that the second marble that is drawn is green, what is the conditional probability that the urn that was selected at the start was A?

10. (Homework.) Assume that we have two coins, one being "fair," that has an equal chance of showing heads and tails when flipped, while the other coin is weighted so that the chance of heads on a flip is 4 / 5 . We pick up one of the coins randomly, then flip it twice. a. What is the probability that both flips show heads?

b. What is the probability that both flips show tails?

c. Given that both flips show heads, what is the conditional probability that the weighted coin was selected?

d. Given that both flips show tails, what is the conditional probability that the fair coin was selected?