## Description

1. (6 points) Suppose the sample space § consists of all seven-letter words having distinct lower

case alphabetic characters.

(a) How many words are there in § ?

(b) How many ”special” words are in § for which only the second, the fourth, and the

sixth characters are vowels, i.e., one of {a, e, i, o, u, y} ?

(c) Assuming the outcomes in § to be equally likely, what is the probability of drawing

such a special word?

2. (4 points) If P(E) = 0.9 and P(F) = 0.9, show that P(EF) ≥ 0.8. In general, prove

Bonferroni’s inequality, namely that

P(EF) ≥ P(E) + P(F) − 1.

3. (6 points) Three balls are selected at random from a bag containing 2 red, 3 green, and 4

blue balls.

(a) What would be an appropriate sample space § ?

(b) What is the the number of outcomes in § ?

(c) What is the probability that all three balls are red?

(d) What is the probability that all three balls are green?

(e) What is the probability that all three balls are blue?

(f) What is the probability of one red, one green, and one blue ball?

2/7 .

4. (4 points) An urn contains four balls, labeled 1 to 4. Balls are drawn at random one by

one, without replacement, until the sum of the numbers on the balls drawn exceeds 4. The

sequence of balls drawn is noted.

(a) (2 points) Write down the sample space for this experiment.

(b) Let E be the event “one of the balls drawn is 1”, and F the event “the final sum of

numbers on the balls drawn is even”. Give the set of outcomes corresponding to each

of the following events

(i) (1 point) “both E and F occur”

(ii) (1 point) “neither E nor F

3)}.

5. (4 points) How many integer solutions are there to the inequality

x1 + x2 + x3 ≤ 17 ,

if we require that

x1 ≥ 1 , x2 ≥ 2 , x

6. (4 points) What is the probability that the sum is less than or equal to 9 in three rolls of a

die?

7. (6 points) Rooks (or castles) are chess pieces that are only allowed to move horizontally or

vertically on a chessboard. Assume that rooks must be placed on one of the 64 allowed

positions on the board, and no two rooks can share the same position.

How many ways are there of placing 8 indistinguishable rooks on a standard 8×8 chessboard:

(a) if there are no restrictions on their placement, other than those stated above?

(b) if no rook can be in a position that attacks another

8. (6 points) A child has 12 blocks: 6 black, 4 red, 1 white, and 1 yellow.

(a) If the child puts the blocks in a line, how many different arrangements are possible?

(b) If one of the arrangements in part (a) is randomly selected, what is the probability that

no two black blocks are next to each other

9. (4 points) A box contains three coins, one of which is fair, one double-headed (i.e., heads on

both sides), and the third is biased in such a way that it comes up heads with probability

3/4. A coin is drawn at random from the box and flipped twice. If both flips result in heads,

what is the probability that the coin drawn was double-headed?

10. (4 points) A monkey at a typewriter types each of the 26 capital letters of the alphabet

exactly once, the order being random.

(a) (2 points) What is the probability that the word ”MONKEY” appears somewhere in

the string of letters?

(b) (2 points) How many independent monkey typists are needed so that the probability

that the word ”MONKEY” appears at least once is greater than 0.9?