Download CM141A – Probability and Statistics I Solutions to exercise

CM141A – Probability and Statistics I
Solutions to exercise sheet 2
1. There are three possible letters that can repeat themselves: R, E and V.
Let’s use the following notation:
R1 = the event that the letter R is chosen from RESERVE
R2 = the event that the letter R is chosen from VERTICAL
and similarly,
E1 = the event that the letter E is chosen from RESERVE
E2 = the event that the letter E is chosen from VERTICAL
V1 = the event that the letter V is chosen from RESERVE
V2 = the event that the letter V is chosen from VERTICAL.
Since the two choices are independent, i.e. choosing letters from the two words does not
influence each other, the probability that the letter R is chosen from both is:
.
Similarly,
.
At the end, we are interested in the union of these (exclusively mutual) events, i.e.
Therefore, all we need now is to calculate the probability for the simple events. For
example,
is the probability of choosing R from RESERVE. The word consists of
7 letters, out of which 2 are R’s. Therefore,
Similarly:
Also:
.
, and
.
.
Therefore,
1
2. Given
and
, and A and B are mutually exclusive events:
(a)
The probability that either A or B occurs:
(b)
The probability that A occurs but B does not is just the probability that A occurs,
since A and B are mutually exclusive events, and so if A happens, B will surely not
happen. Put up in formulas:
(c)
The probability that both A and B occur is just
by definition, since
they are mutually exclusive!!!
3. Let’s use the notation for the following three events:
S – a student attends the Spanish class
F – a student attends the French class
G – a student attends the German class
And let us summarize the information given in the problem in a probabilistic way:
,
and
.
,
And last,
(a)
and
.
.
The event that a student is not in any of the language classes is just
(The shaded area in the following Venn diagram)
F
S
G
And thus its assigned probability is:
where
2
(b)
The event that a student is taking exactly one language class is
And thus its assigned probability is:
Note that
Similarly
And thus
(c)
If 2 students are chosen randomly, what is the probability that at least one is
taking a language class?
Solution: The event that at least one student is taking a language class is the complement
of no student taking a language class. Therefore, it is useful to focus on the latter.
The sample space of choosing 2 students at random is
,
while the number of outcomes where no student takes a language class is
2 students out of the 50 that does not take language classes)
Therefore, the probability that the two students take no language classes is
,
and the required probability is just
.
4. There are
3
(choosing
ways of dividing the 40 players into 20 ordered pairs of two each. (That is, there are
ways of dividing the players into a first pair, a second pair, and so on.) Hence
there are
ways of dividing the players into (unordered) pairs of 2 each.
Furthermore, since a division will result in no offensive-defensive pairs if the offensive
(and defensive) players are paired among themselves, it follows that there are
such divisions. Hence the probability of no offensive-defensive
roommate pairs, call it
, is given by
.
To determine
that there are
, the probability that there are 2i offensive-defensive pairs, we first note
ways of selecting the 2i offenrive players and the 2i defensive
players who are to be in the offensive-defensive pairs. These 4i players can then be paired
up into (2i)! possible offensive-defensive pairs. (This is so because the first offensive can
be paired with any of the 2i defensives, the second offensive with any of the remaining
defensives, and so on.) As the remaining
offensives (and defensives) must
be paired among themselves; it follows that there are
divisions which lead to 2i offensive-defensive pairs. Hence
.
The
can now be computed or they can be approximated by making use
of a result of Stirling which shows that n! can be approximated by
For instance, we obtain that
4
.
.
5. In order to answer these questions it is useful to calculate certain conditional
probabilities. We know that if
or if
then the
events are independent.
a. From the table:
Thus,
and the events are not independent
b. From the table:
Thus,
and the events are independent!
c. From the table:
Thus,
and the events are not independent
As for events C and E, note that the event E is the complement of event D
know from the lectures that if C is independent of D then C is independent of
. We
,
and so by contradiction C and E must be dependent.
6. Define the events:
A:
The offender has 10 or more years of education.
B:
The offender is convicted within two years after completion of treatment.
a. Directly from the table
.
b. Directly from the table
.
c. Directly from the table
d. By using the law of addition we obtain:
e.
f.
5
g.
h. By definition of the conditional probability
i. By definition of the conditional probability
7. Note that by definition of the conditional probability
, where here
. By using the law of addition (for the mutually exclusive events A and B) we obtain:
And it is also clear that
Thus in summary:
. QED
8. A and B are two events such that
and
In case the events A and B are mutually exclusive, then
, and
that value. On the other hand
exceeds
and any value
smaller than that is allowed (Draw a Venn diagram to convince yourself). Therefore, in
full generality
a. Is it possible that
, and the answers are:
? Possible.
b. What is the smallest possible value of
c. Is it possible that
?0
? Why or why not? Impossible, as it exceeds 0.3
d. What is the largest possible value of
? 0.3
9. A and B are independent events,
, and
A and B are independent events, and therefore
6
.
Also:
Solving this equation for
gives
.
10. Define the events:
A:
The person is traveling on business.
C1:
The person is flying on a major airline.
C2:
The person is flying on a privately owned plane.
C3:
The person is flying on a commercially owned plane not belonging to a
major airline.
And let us summarize the information given in the problem using formulas:
•
Of the travelers arriving at a small airport, 60% fly on major airlines:
•
30% fly on privately owned planes:
•
The reminder flies on commercially owned planes not belonging to a major
airline:
•
Of those traveling on major airlines, 50% are traveling for business reasons:
•
60% of those arriving on private planes are traveling for business reasons:
•
90% of those arriving on other commercially owned planes are traveling for
business reasons:
.
Suppose that we randomly select one person arriving at this airport. What is the
probability that the person
a. is traveling on business?
b. is traveling for business on a privately owned plane?
7
c. arrived on a privately owned plane, given that the person is traveling for business reasons?
d. is traveling on business, given that the person is flying on a commercially owned plane?
Note: here we are assuming an “other” commercially owned plane, otherwise we need to
include the major airlines as well (since they are also, strictly speaking, commercially
owned).
In problems on probability, there are often confusions of this type regarding the precise
phrasing of the problem. If something confuses you, please write out your assumptions
carefully.
8