Question 1:

A hot dog vendor provides onions, relish, mustard, catsup, and hot peppers for your hot dog. Determine how many combinations of toppings (no "double" toppings) are possible assuming:

(a) You use exactly one topping.
(b) You use exactly two toppings.
(c) There are no restrictions on how many toppings you can use.

Question 2:

There are in teams in league A and n teams in league B. On game day there are k games played (k ≤ n and k ≤ m); each game pits a team from league A against a team from league B. A given team can play at most once on game day. How many different sets of match-ups ("schedules") can be made for game day?

(a) If a "schedule" includes the teams playing and a distinct time for each of the k games?

(b) If a "schedule" includes only the team match-ups and no information on time?

Question 3:

Consider the set of 4-tuples (w, x, y, z) where w, x, y, and z are non-negative integers and

w + x + y + z = 32

If a member of this set is drawn at random, what is the probability that the value of w in the drawn 4-tuple is 4?

Question 4:

Population Estimation: It is desired to estimate the number of foxes in a forest without catching all of them. Previously, 10 foxes have been captured, tagged and released. A month later 20 foxes are captured and 5 of these have tags.

(a) If the actual number of foxes in the forest is N (unknown to us), what is the probability of this event as a function of N? Denote this probability by p(N).

(b) The estimate of the number of foxes is taken to be the value of N which maximizes p(N). What is this estimate?

Question 5:

A study is being conducted in an attempt to correlate quality of education and income level. Engineers who have been in the workforce for at least ten years are categorized according to their income level being one of HIGH, MEDIUM, or LOW. Their undergraduate colleges are also noted and this population is limited to those who graduated from USC, Stanford, or Harvey Mudd College (HMC). Half of this population graduated from USC, ten percent from HMC and the rest graduated from Stanford. For USC graduates, 60% are HIGH income earners and 30% are MEDIUM income earners. Half of Stanford graduates earn HIGH income and 20% earn LOW income. HMC graduates make the most; 90% of HMC graduates earn high income and the rest are equally likely to be LOW or MEDIUM wage earners.

(a) What is the probability that an engineer is an HMC graduate and makes a HIGH salary?

(b) What is the probability that an engineer is an HMC graduate and does not make a LOW salary?

(c) Determine the probability that an engineer has a LOW, MEDIUM, or HIGH income:

i. P(LOW income)
ii. P(MEDIUM income)
iii. P(HIGH income)

(d) If an engineer has a LOW income, what is the probability that she graduated from Stanford?

(e) Suppose that you're given that a particular engineer has a HIGH income and asked to make your best decision as to which school she attended. Describe a strategy for making this decision and then use your strategy to make your best decision.

Question 6: Consider the probability space associated with rolling a fair die with sample space u = {1, 2, 3, 4, 5, 6}. Provide an example of two non-empty events associated with this experiment, A and B, that are mutually exclusive. Provide an example of two non-empty events associated with this experiment, C and D, that are statistically independent.

Question 7:

Consider three events A, B, and C with set relations show in Fig. Give the simplest expression for the following probabilities:

(a) P(A U B U C)

(b) P(A ∩ B ∩ c)

(c) P(A| B U C)

(d) P(A|B ∩ C)

(e) P(B U C|A)