Rensselaer Polytechnic Institute
Department of Electrical, Computer, and Systems Engineering
ECSE 2500: Engineering Probability, Fall 2011
Exam 1 Solutions
The boxed numbers beside each problem give the mean (left) and median (right) score for each
problem. If your score on a problem is lower than the median, you need to learn the concept better.
The overall mean was 78.1 and the median was 83.5. There were 18 scores in the 90’s.
12.8
13.5
1. (15 points.) In this problem we consider two fair 4-sided dice. Die A is labeled with the numbers
{
1
,
1
,
2
,
3
}
and Die B is labeled with the numbers
{
0
,
1
,
1
,
2
}
. The two dice are rolled and we record
the two results.
4.4
5
(a) (5 points.) What is the simplest and smallest sample space
S
we can use for this problem? How
many possible outcomes are there?
Since there are three outcomes for each die, the sample space
S
for the two results (die A, die
B) has nine total outcomes:
S
=
±²
1
0
³
,
²
1
1
³
,
²
1
2
³
,
²
2
0
³
,
²
2
1
³
,
²
2
2
³
,
²
3
0
³
,
²
3
1
³
,
²
3
2
³´
Since we cannot distinguish the 1’s on Die A and B, it is not correct to say we have 16 outcomes,
or that we have some sort of expanded sample space with repetitions. Also, since we record the
two results separately, the sample space is not the sum of the outcomes.
It is critical to see that these outcomes are not all equally probable! For example,
±²
1
1
³´
is
four times as likely as
±²
3
2
³´
.
4.4
5
(b) (5 points.) What is the event
A
corresponding to “The two results sum to 3”?
Three of the above outcomes are in the event
A
:
A
=
±²
1
2
³
,
²
2
1
³
,
²
3
0
³´
The event must be an explicit subset (a collection of outcomes) from the sample space
S
in part
(a). While it wasn’t part of the question,
P
(
A
) =
5
16
.
1