Stat
201
Exam
2
Spring
2008
Version 2 KEY
Page
1
of
9
1.
It was recently announced that, In the United States, 1% of U.S. adults overall are
currently in prison. However, among just blacks, 7% of U.S. black adults are currently in
prison.
this is P(prison GIVEN black)
i.e., P(Y | X) = .07
Suppose you randomly select one U.S. adult.
Let
X
represent the event ‘the adult is black’.
Let
Y
represent the event ‘the adult is currently in prison.’
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) No, because 1%
7%.
(ii) Yes, because 1%
7%.
(iii) No, because the adult could be both black and in prison.
(iv) Yes, because the adult could be both black and in prison.
(v) No, because only one adult is selected.
(vi) Yes, because only one adult is selected.
It is certainly possible for the adult to
be black, and to also be incarcerated.
(b)
Are the two events independent? Circle the best answer:
[3 pts]
(i) No, because 1%
7%.
(ii) Yes, because 1%
7%.
(iii) No, because the adult could be both black and in prison.
(iv) Yes, because the adult could be both black and in prison.
(v) No, because only one adult is selected.
(vi) Yes, because only one adult is selected.
With one randomly selected person,
P(prison) = P(Y) = .01
but
P(prison GIVEN black) = P(Y | X) = .07.
Since these are not the same, ‘being
black’ is not independent of ‘being in
prison’.
(c)
Suppose that you first look at the adult’s race, but not yet at their incarceration
status. If you find that the adult is black, then,
given
that fact, which of the
following is the probability that they will also be in prison?
[3 pts]
The stated 7% is
P(prison GIVEN black)
(i)
.
07
(iv)
.
08
(ii)
.
01
(v)
.
007
(iii) zero
(vi) close to 100%