Part A

1. The blood types of a group of 3 people are O, A, and B, respectively. Suppose two people are randomly selected with replacement (that is, a person is selected and his/her blood type is observed. This person then returns to the group). Let A be the event of selecting a person of type A, B be the event of selecting a person of type B, and O be the event of selecting a person of type 0.

a) List all possible outcomes. 

b) List all possible outcomes for each of the following events and find the corresponding probabilities.

i. E1 = {Exactly 1 person of type A is drawn}

ii. E2 = {The first person has type B}

iii. E3 = {At least one person has type 0}

iv. E4 = {Both people have the same blood type} List all possible outcomes and find the probabilities of the following events.

c) List all possible outcomes and find the probabilities of the following events.

i. Not E3

ii. E1 & E3

iii. E1 & E4

iv. E2 & E4

V. E3 or E4

d) Identify all possible pairs of events defined in part (b) that are mutually exclusive.

e) Verify mathematically that E2 and E4 are independent events, while E3 and E4 are not.

2. Suppose we select a random sample of 3 numbers between 1 and 40, sampling without replacement.

a) How many samples of size 3 are possible?

b) What is the probability that the sum of the values in our sample is less than 11?

3. There are four ABO blood types: A, B, AB and O. Your blood type is also determined by RH status: Rh+ or Rh-. That leaves us with eight possible blood types: A+, A-, B+, B-, AB+, AB-, 0+, 0-. Among a group of 50 patients, 18 of them have type A+, 3 have type A-, 4 have B+, 2 have B-, 4 have AB+, 2 has AB-, 13 have type 0+, and 4 have type O-. A random sample of 8 patients will be selected from this group.

a) How many different samples are possible?

b) How many different samples of size 8 are possible subject to the constraint that no 2 patients may have the same blood type?

c) What is the probability that a random sample of 8 patients from this group has no two patients with the same blood type?

4. Assume that a 25-year-old man has these probabilities of dying during the next five years:

a) What is the probability that the man does not die in the next five years?

b) An online insurance site offers a term insurance policy that will pay $100,000 if a 25-year-old man dies within the next five years. The cost is $175 per year. So the insurance company will take in $875 from this policy if the man does not die within five years. If he does die, the company must pay $100,000. Let X denote the cash intake of the insurance company which depends on how many premiums the man paid. Find the distribution of X.

Hint: if a 25-year-old man dies in the first year, the cash intake of the insurance company is -(100,000-175)=-$99,825 under the policy; if he dies in the second year, the cash intake of the company is -(100,000-2*175)=-$99,650 etc. If the man does not die within five years, the cash intake of the company is 5*175=875.

c) What is the insurance company's mean cash intake?

d) Suppose the insurance company insures one hundreds 25-year-old men under the terms of Question b). What is the probability that the insurance company will receive at least one claim? For simplicity, assume independence.

5. Three friends are trying to predict who will make more money at work tomorrow. John works construction: he will make $250 tomorrow unless it rains, in which case he will make $0. Sara works in a restaurant: she will earn $100 if it is sunny, $200 if it is rainy, and $250 if the weather is average. Chris works in a warehouse, and he earns a flat rate of $200 per day, regardless of weather conditions. Suppose there is a 20% chance that tomorrow will be sunny, and 50% chance that the weather will be average, and a 30% chance of rain.

a) Determine tomorrow's expected earnings for each of these 3 friends.

b) Compute the standard deviation for Sara's earnings.

6. Courtney is a basketball player who makes 65% of her free throws. She attempts 10 free throws.

a) What is the probability that she will miss more than 8 of them?

b) What is the probability that she will miss at least 2 of them?

c) How many throws do you expect Courtney will miss?

d) Obtain the standard deviation of the number of throws that Courtney misses.