**Econ 57 Fall 2001
Midterm Answers**

**1**. The probability
of five heads is(1/2)^{5} = 1/32; the probability of five tails is (1/2)^{5}
= 1/32. Therefore, the probability of five heads or five tails is 2/32 = 0.0625.

**2**. Using Bayes’ Rule,

**3**. All three cases can be
answered by seeing which city has the SMALLER standardized Z value:

a. ZA = (100 - 80)/20 = 1, ZB = (100 - 80)/30 = 2/3. So, B.

b. ZA = (100 - 80)/20 = 1, ZB = (100 - 70)/20 = 3/2. So, A.

c. ZA = (100 - 80)/20 = 1, ZB = (100 - 70)/30 = 1. So, equal.

**4**. The total area of the
bars is not 1.

**5**. Because Mark has a higher
probability of snow, he should bet on snow and Mindy should bet against snow.
So, we will make the bet that Mindy pays Mark $X if it snows and Mark pays Mindy
$Y if it doesn’t.

From Mark’s viewpoint, his expected value is (X)(4/5) + (-Y)(1/5). For this to be positive, he needs X/Y > 1/4. From Mindy’s viewpoint, her expected value is (-X)(2/3) + (Y)(1/3). For this to be positive, she needs X/Y < 1/2. Thus, any bet with 1/4 < X/Y < 1/2 will give both persons positive expected values. For example, Mindy pays Mark $1 if it snows and Mark pays Mindy $3 if it doesn’t.

**6**. Here are the answers:

a. scatter diagram

b. scatter diagram

c. side-by-side boxplots

d. time series graph

e. scatter diagram

**7**. Marilyn’s correct
answer: “I’d choose heads/tails (HT), because I’d win three out
of four times! That’s because there are four different sequence combinations:
HH, HT, TH, and TT. If tails/tails (TT) were to appear at the very start, you’d
win, but that would happen only one-fourth of the time. For TT to appear any
time afterward, it would have to be preceded by H. Which means that I’d
win before you ever saw your sequence come up at all!”

**8**. Using the binomial distribution
with p = 0.2 and n = 5:

a. P[X = 5] = 0.2

^{5}.

b. Similarly,

**9**. The mean is 2.3, the median
is 2, the standard deviation is 1.

**10**. Marilyn’s (correct)
answer [“Ask Marilyn,” Parade, October 29, 2000.]:

The distribution of sexes will remain roughly equal. That’s because—no matter how many or how few children are born anywhere, anytime, with or without restriction—half will be boys and half will be girls! Only the act of conception (not the government!) determines the sex.

One can demonstrate this mathematically. (In this case, we’ll assume that women with firstborn girls will always have a second child.) Let’s say 100 women give birth, half to boys and half to girls, The half with boys must end their families. There are now 50 boys and 50 girls. The half with girls (50) give birth again, half to boys and half to girls. This adds 25 boys and 25 girls, so there are now 75 boys and 75 girls. Now all must end their families. So the result of the policy is that there will be fewer children in number, but the boy/girl ratio will not be affected.