1. [15 pts total] Amazon sends 40% of its overnight mail parcels via express mail service E1. Of these parcels, 2% arrive after the guaranteed delivery time (denote the event "late delivery" by L). 50% of overnight parcels are sent via express mail service E2. Of those sent via E2, only 1% arrive late. The remaining 10% of overnight parcels are sent via E3. Of those sent via E3, 5% of the parcels arrive late.

(a) [5 pts] Draw a probability tree that describes this experiment. Be sure to write the joint probabilities at the end of each branch.

(b) [1 pts] If a record of an overnight mailing is randomly selected from the company's file, what is the probability that the parcel went via E1 and was late?

(c) [4 pts] What is the probability that a randomly selected parcel arrived late

(d) [5 pts] If a randomly selected parcel has arrived on time, what is the probability that it was not sent via E1?

2. [15 pts total] One percent of all individuals in a certain population are carriers of a particular disease. A diagnostic test for this disease has a 90% detection rate for carriers and a 5% detection rate for non-carriers. Suppose the test is applied independently to two different blood samples from the same randomly selected individual.

(a) [6 pts] Draw a probability tree that describes this experiment. Be sure to carefully define your notation and write the joint probabilities at the end of each branch.

(b) [6 pts] What is the probability that both tests yield the same result?

(c) [3 pts] If both tests are positive, what is the probability that the selected individual is a carrier?

3. [20 pts total] Answer the following short questions. Show all your work where possible (if the answer is incorrect, part marks will be based on work shown).

(a) [5 pts] Suppose that the probability of living to be older than 70 is .6 and the probability of living to be older than 80 is .2. If a person reaches her 70th birthday, what is the probability that she will live beyond her 80th birthday?

(b) [5 pts] Is it true in general that P(A)P(A \ B) (P(A))2. Explain your answer carefully.

(c) [5 pts] In an experiment with two independent fair coin tosses, let A be the event of observing two heads and let B be the event of observing two tails. Are the events A and B independent? Justify your answer using probability. [Hint: write out full the sample space to find the required probabilities]

(d) [5 pts] In an experiment with two independent fair coin tosses, let A be the event of observing heads on the first toss and let B be the event of observing heads on second toss. Are the events A and B independent? Justify your answer using probability. [Hint: write out full the sample space to find the required probabilities]

4. [30 pts total] Solve the following combinatorial problems. Show all your work where possible (if the answer is incorrect, part marks will be based on work shown).

(a) [5 pts] Beethoven wrote 9 symphonies, Mozart wrote 27 piano concertos, and Schubert wrote 15 string quartets. A university radio announcer decides that on each successive night, a Beethoven symphony will be played, followed by a Mozart piano concerto, followed by Shubert string quartet. What is the maximum number of days that she can continue this practice before having to repeat a program (i.e., a selection of symphony, concerto and string quartet)?

(b) [5 pts] A chemist is studying the effects of temperature, pressure, and type of catalyst on a chemical reaction. The chemist can set the temperature to one of 3 different levels, the pressure to 4 different levels, and can choose among 5 catalysts. How many possible experiments can be conducted in which either the lowest temperature or one of the two lowest pressures are used? [Note that this includes the cases in which the lowest temperature and one of the two lowest pressures are used together.]

(c) [5 pts] Three married couples have purchased theater tickets and are seated in a row consisting of just six seats. If they take their seats in a completely random fashion (random order), how many ways are there for husband and wife Jim and Paula to sit next to one another?

(d) [5 pts] Our class has 50 students. Assuming that no students are born on leap days (February 29), what is the probability that no two students share the same birthday? What is the probability that at least one of the students has the same birthday as another student in the class? Please provide your answers in the form of a fraction.

(e) [5 pts] Five cards are dealt from a standard 52-card deck. We define a "straight" hand as one that consists of five cards where all the five cards are consecutive (regardless of their suit). Note that the order of the cards are Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King and that a "straight" is not cyclic (for example 9, 10, J, K, Q is a straight but J, Q, K, A, 2 is not). Find the probability of the following events:

i. A straight consisting of 1 Ace, 1 two, 1 three, 1 four, 1 five.

ii. A hand consisting of 1 Ace, 1 three, 1 five, 1 seven, 1 nine.

iii. Any straight as defined in the problem statement.

(f) [5 pts] In a jar are 5 red marbles, 10 blue marbles, and 5 green marbles. I draw three marbles from the jar (without replacement). What is the probability that at least two of these marbles are of the same color?

5. [20 pts total] Solve the following problems. For this question you must derive probability mass functions from first principles (e.g. using counting rules and properties of probabilities) instead of attempting to use the "named" distributions that we will discuss next week. Note: when writing down probability mass functions, the range of the variable must always be clearly provided.

(a) [5 pts] Tay-Sachs disease is a rare but fatal disease of genetic origin occurring mostly in infants and children, especially those of Jewish or eastern European descent. If a couple are both carriers of Tay-Sachs disease, a child of theirs has probability .25 of being born with the disease. If such a couple has four children, what is the probability mass function for the number of children who will have the disease? What important assumption did you make? Please define all notation, state any assumptions, and clearly state the probability mass function as a formula.

(b) [5 pts] My spam filter flags 2% of (my) emails, which I assume arrive in my inbox at random. Let the random variable Y be the number of emails I receive until I see the second spam message (including the second spam message). Write down the probability mass function of Y as a formula. [Hint: write down p(y) for a few cases and then try to guess a formula for a general y from the pattern]

(c) [5 pts] A jar contains 5 blue and 10 green marbles. You draw 3 marbles out of the jar without replacement. Let Y be the number of the drawn marbles that are blue. Write down the probability mass function of Y as a formula.

(d) [5 pts] Suppose that a random variable Y takes only the following values: y= 1,2,3,4. The

probability mass function for the three values y = {1, 2, 3} is p(y) = (y^2 + 4y 1)/40,

y = 1, 2, 3. However, the probability that Y = 4 is not provided. Find P(Y = 4).