Economics 521 Fall, 2007 First exam.
Do all five problems and the sixth if you choose. EXPLAIN your reasoning for
each answer!! [Tell me WHY you add when you add, WHY you multiply when
you multiply, ... .] Include a statement of exactly what assumptions you are
making.
1. (20 pts) A. (15 pts) You have in your closet 10 different pairs of shoes for a
total of 20 shoes. Each morning Professor Evensky goes to your closet and
randomly selects six single shoes from your closet. What is the probability that
in that set of six shoes there is at least one pair?
B. (5 pts) Do the problem again, but this time assume Professor Evensky
randomly selects eleven shoes.
2. (20 pts) Consider a fair die that we are going to toss 20 independent times.
(A) Which of the following sequences is more likely
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
or
1 4 3 6 4 5 4 1 2 4 2 3 6 6 2 1 4 3 2 5
(B) Which is more likely, getting three 1s in 20 tosses or getting four 1s?
3. (20 pts) On an assembly line, 50% of all widgets are put together by Mary,
40% by Bill, and 10% by Nicole. 5% of the widgets Mary assembles malfunction
compared with 10% of those assembled by Bill and 20% by Nicole. If a widget
malfunctions, what is the probability that it is put together by Mary? by Bill? by
Nicole?
4. (20 pts) For the experiment of dealing a five card hand from a well-shuffled
poker deck, the random variable X is the number of denominations represented
in the hand. (Recall there are 13 denominations altogether: 2, 3,..., Q, K, A.)
Determine the probability distribution of X (16 pts) and calculate the mean of
this random variable (4 pts).
5. (20 pts) Raoul tosses a fair coin until the first heads turns up, then stops;
Maria independently also tosses a fair coin until the heads turns up and then
stops. What is the probability that they will stop after the same number of
tosses? [So, if Raoul gets TTTH and Maria gets TTH, they do not have the same
number; but if both get TTTH, they do have the same.]
6. (extra credit general culture question) Who is the current world record
holder for fastest time running a marathon?