Probabilty

1. You roll two dice.

a) What is the probability that the sum of the dice is eight?

b) What is the probability that the first die is a two?

c) What is the probability that the sum is 8 and the first die is 2?

d) What is the probability that the sum is 8 or the first die is 2?

(e) Are these events (“sum is eight” and “1st is two”) independent? Explain

(f) Are these events (“sum is eight” and “1st is two”) mutually exclusive? Explain

2. In a group of 200 voters, there were 80 Republicans, 50 voters over the age of fifty-five, and 10 of the Republicans were over fifty-five. If a voter is chosen at random, find:

(a) p(voter is Republican)

(b) p(voter is over fifty-five)

(c) p(voter is Republican OR over fifty-five)

(d) Are these events (“voter is Republican” and “voter is over 55”) independent? Explain

(e) Are these events (“voter is Republican” and “voter is over 55”) mutually exclusive? Explain

Practice with Expected Value

1. A manufacturer of telephones realizes a profit of $5 for each telephone sold. However, defective phones can not be sold and cost $10 to produce. Find the expected profit if the probability that a phone is defective is 2%.

2. Insurance companies use probability and expected value to determine their rates. Suppose that 200,000 out of 1,000,000 drivers between the ages of 18 and 22 had a car accident. If the average accident cost the insurance company $1,500 and the insurance company charges $800 for an insurance policy, will the company make a profit?

3. A lawyer sometimes represents client for a contingency fee. The lawyer only gets paid for services rendered if the client wins the case. Suppose a client is suing someone for $400,000 and the lawyer's fee is 10% of the settlement. The lawyer will spend $2,000 preparing the case and believes there is a 20% chance of winning the case. Find the lawyer's expected profit.

4. In Atlantic City, a card game known as blackjack or “21” is played. Originally, everyone gets two cards. One of the dealer's cards is face up and the other is face down. If the dealer's face up card is an ace, you can bet $10 that the dealer's other card is a face card (10, J, Q, K). If the card is a face card, you are paid $15 (in addition to the return of your $10). If it is not a face card, you lose the $10. What is the expected profit from the bet?

5. Find the mean (expected value) and the standard deviation for the distribution of the difference (absolute value) of two dice.

Practice with the Binomial Distribution

1. If a couple has five children, what is the probability that :

(a)  they have three boys and two girls (b) all five are boys (c) at least one is a girl (d) they have more than 3 boys

2.  A drug company claims that 10% of patients who use their drug experience side effects. If a group of eight patients is given the drug, what is the probability that:

(a)  more than one will experience side effects

(b)  no patients experience side effects.

(c) What is the expected number of patients who experience side effects in the group of eight?

(d) What is the standard deviation of the number of patients experience side effects for groups of eight patients?

(e) Suppose 7 out of the 8 patients experience side effects. What would you conclude?

3.  In a redundant communications network, there are 6 paths between two nodes, so that if a message can not get through one path, it can be sent through one of the other paths. Suppose that the probability that an individual path is busy is 80%. What is the probability that a message can be sent from one node to the other?

4.  A multiple choice test consists of 6 questions; each question has answer choices A, B, C, D, and E. What is the probability of guessing:

(a)  all six correct answers

(b)  at least 5 correct answers

(c)  at least one correct answer

(d)  Find the expected number of correct answers and the standard deviation of the number of correct answers

5.  A company has determined that 1% of the parts it makes are defective. If the company packs a shipment of 5 parts to a customer, what is the probability that: (a) none are defective (b) more than 2 are defective.

6.  A basketball player has an average of making 80% of her free throw attempts. What is the probability that the player makes 4 out of 6 attempts today? What is the mean and s.d. for the number made in all six attempt games?

7. Describe in a short paragraph the relationship between independent events and mutually exclusive events.Your answer should include (but should not be limited to) an answer to the following questions:

•Give an example of two events A and B that are mutually exclusive, and explain why

they are dependent events.

•Give an example of two events that are dependent, but are not mutually exclusive.

Both examples and explanations are desired. Most important here is that you understand the concept of independent events, it plays an important role in statistics!