Question

1. A baseball team has a roster of 25 players. Call these 25 players the roster. Only 9 of the players from the roster are actually on the field at a given time. Call these 9 players the team.

(a) How many combinations of teams are there if any player can play any position?

(b) How many permutations of teams there if any player can play any position?

(c) Suppose the roster has 2 catchers, and 7 pitchers. At any time the team can have only one catcher and one pitcher. How many combinations of teams are there in this case?

2. . A bowl contains 12 marbles, of which 5 are red, 4 are blue and 3 are white. Three marbles are drawn at random from the bowl:

(a) Compute the probability of drawing 3 white marbles if the marbles are drawn with replacement. That is you place a marble back in the bowl after you draw it.

(b) Compute the probability of drawing 3 white marbles if the marbles are drawn without replacement. That is you do not place a marble back in the bowl after you draw it.

(c) Compute the probability of drawing 3 marbles of different color (one red, one blue, one white) with and without replacement.

3. What is the probability of drawing a full house (two of a kind + 3 of a kind) in a hand of five cards from a deck of 52 cards?

4. From a regular deck of 52 playing cards, cards are drawn successively and without replacement. Compute the probability that the third spade appears on the fifth draw.

5. A survey organization asked respondents from 3 different geographical regions which of 3 presidential candidates they preferred. The answer are reported below:

	East	Midwest	West
A	120	80	110
B	15	65	45
C	30	40	15
Total	165	185	170

(a) What is the probability that a randomly selected respondent from this sample prefers candidate A?

(b) What is the conditional probability that a respondent prefers candidate A, given that the respondent comes from the Midwest?

(c) What is the probability that a respondent in the sample prefers candidate B and comes from the East?

(d) What is the conditional probability that given that a respondent is from the East, given that this respondent prefers candidate A?

Answer 1