201
Exam
2
Spring
2008
Version 1 KEY
Page
1
of
9
1.
In the U.S., 12% of the population is black.
Suppose, of all U.S. families that have a brother and a sister, you randomly select one
such family.
Let
X
represent the event ‘the brother is black’.
Let
Y
represent the event ‘the sister is black.’
[Note that some U.S. families have a child who is adopted (but most don’t).]
Remark: This exercise was anticipated by exercises 1 and 2 on the practice exam; it is
just like scenario 3 of lab 6 and exercise 1 of lecture 12; the ideas were covered in various
exercises of homework 6, and throughout lectures 10, 11, and 12.
(a)
Are the two events disjoint? Circle the best answer:
[3 pts]
(i) Yes, because both siblings could be black.
(ii) No, because both siblings could be black.
(iii) Yes, because each sibling is selected from the same family.
(iv) No, because each sibling is selected from the same family.
(v) Yes, because there’s a 12% chance the parents are black.
(vi) No, because there’s a 12% chance the parents are black.
It is certainly possible for the brother to be
black, and for the sister to also be black.
In fact, since about 12% of families are black,
there’s about a 12% chance of finding a black
brother along with a black sister. Symbolically,
P(X and Y) = about 12%.
Since P(X and Y)
0, the events aren’t disjoint.
(b)
Are the two events independent? Circle the best answer:
[3 pts]
(i) Yes, because both siblings could be black.
(ii) No, because both siblings could be black.
(iii) Yes, because each sibling is selected from the same family.
(iv) No, because each sibling is selected from the same family.
(v) Yes, because there’s a 12% chance the parents are black.
(vi) No, because there’s a 12% chance the parents are black.
If the first sibling is black, then most likely
you’ve selected a black family, in which case
there would be a nearly 100% chance the other
sibling is black; whereas, if the first sibling is
not black, then you’ve most likely selected a
non-black family, in which case there is a nearly
zero chance the other sibling is black.
So, knowing the outcome of X changes your
estimated likelihood of Y.
Symbolically, P(Y) = .12, but P(Y given X) =
nearly 100%
(c)
Suppose that you first look at the brother, but not yet at the sister. If you find that
the brother is black, then,
given
that fact, which of the following is most
reasonably the probability that the sister will then be black?
[3 pts]
(i)
zero
(iv)
.
144
(ii)
.
12
(v)
.
096
(iii)
.
24
(vi) close to 100%
If the first sibling is black, you have
most likely selected a black family.
The only reason the other sibling
might not also be black is if one of
the
siblings
was
adopted
(or
if
perhaps the mother had the siblings
with different fathers of different
races), either of which is a very
unlikely scenario which only happen