Solutions for STAT 101, Homework 5
Professor Rakhlin
Solutions prepared by Julie, Tung, and Wei
October 23, 2011
Problem 1
While a regular coin is unlikely to land on its side, one can glue together several coins and thus
create a “threesided coin that has equal probability of landing “Heads, “Tails, and “Side.
a) Suppose we toss such a threesided coin twice. What is the best sample space S? Are outcomes in this
space equally likely? What is the probability that the threesided coin lands on its “Side at least once?
b) Suppose we toss the threesided coin three times. Is getting HeadsHeadsHeads more likely than getting
HeadsTailSide?
Answer:
a) (4 points) For simplicity, denote ”Heads” as ”H”, ”Tails” as ”T”, and ”Side” as ”S”. The best sample
space is the only sample space: S =
{
HH,HT,HS, TT, TH, TS, SS, SH, ST
}
.
Since P(H) = P(T) = P(S) and each toss is independent, each outcome is equally likely.
P
(
S at least once of two tosses
) =
{
HS,TS,SS,SH,ST
}
{
HH,HT,HS,TT,TH,TS,SS,SH,ST
}
=
5
9
Student don’t need to explain the details, as long as they get the correct answers. One point for the
correct sample space. One point for stating the outcome is equally likely and 2 points for the calculation.
b) (1 point) No, all outcomes are equally likely since the coin is fair ( the probabilities of getting a head, a
tail, or a side are the same).
Problem 2
A club has 100 members. Among them there are 40 lawyers and 50 liars. The number of mem
bers who are neither lawyers nor liars is 20. A club president is chosen from the 100 members at random.
a) What is the probability that the club president is a lawyer?
b) What is the probability that the club president is a lawyer and a liar?
c) If you know that the president is a lawyer, what is the probability that he is also a liar?
Answer:
a) (1point) Let
A
be the set of lawyers and
B
be the set of liars.
P
(president is a lawyer) =
40
100
=
2
5
= 0
.
4.
b) (2 points)
A
∩
B
is the set of those who are both lawyers and liars.

A
∩
B

=

A

+

B

A
∪
B

. Since there
are 20 people who are neither lawyers nor liars,

A
∪
B

= 100

20 = 80. So,

A
∩
B

= 40 + 50

80 = 10.
P
(club president is a lawyer and a liar) =
10
100
=
1
10
= 0
.
1
c) (1 point) This is a conditional probability of the president being a liar given that he is a lawyer.
P
(president is liar

president is a lawyer) =
P
(president is both a liar and a lawyer)
P
(president is a lawyer)
=
0
.
1
0
.
4
=
1
4
= 0
.
25
Problem 3
It is generally thought that athletic kids are more popular than unathletic kids; at least in Middle
School. Is this true for Penn Students too? Penn Students were asked questions about their popularity
and athleticism. The data table PennPopularity.jmp records each students selfreported Sex and level of
1