4CCM141A/5CCM141B – Probability and Statistics I
Problem Set 1
1. If there are 12 strangers in a room, what is the probability that no two of them celebrate
their birthday in the same month?
2. A vehicle arriving at an intersection can turn right, turn left, or continue straight ahead.
The experiment consists of observing the movement of a single vehicle through the
intersection.
a) List the sample space of this experiment.
b) Assuming that all the sample points are equally likely, find the probability that the
vehicle turns.
Another experiment consists of observing two vehicles moving through the intersection.
c) How many sample points are there in this experiment? List them.
Assuming that all sample points are equally likely,
d) what is the probability that at least one car turns left?
e) what is the probability that at most one car turns?
3. A fleet of nine taxis is to be dispatched to three airports in such a way that three go to
airport A, five go to airport B, and one goes to airport C. In how many distinct ways can
this be accomplished?
Assume that the taxis are allocated to the airports at random.
a) If exactly one of the taxis is in the need of repair, what is the probability that it is
dispatched to airport C?
b) If exactly three of the taxis are in the need of repair, what is the probability that every
airport received one of the taxis requiring repairs?
4. (a) Ten children are to be divided into an A team and a B team of 5 each. The A team
will play in one league and the B team in another. How many divisions are possible?
(b) In order to play a game of basketball, 10 children at a playground divide themselves
into two teams of 5 each. How many different divisions are possible? (Note that this is
different from (a)).
5.
(a) Consider the binomial expansion
n
⎛n⎞
n
x
+
y
=
(
) ∑ ⎜ ⎟ x i y n −i
i =0 ⎝ i ⎠
Provide a combinatorial proof of this theorem.
Hint: consider a more general case ( x1 + y1 )( x2 + y2 )L ( xn + yn ) , i.e. when one adds
subscripts that distinguish between the different x’s and y’s. A straightforward expansion
yields 2n terms, some of which turn out to be identical once the indices are removed.
One needs to count exactly number of different terms that contribute to xi y n −i .
6. (a) A balanced die is tossed six times, and the number on the uppermost face is
recorded each time. What is the probability that the numbers recorded are 1,2,3,4,5,6 in
any order?
(b) Suppose that the die has been altered so that the faces are 1,2,3,4,5,5 (i.e. 5 appears
twice and there is not any 6). If the die is tossed five times, what is the probability that
the numbers recorded are 1,2,3,4,5 in any order?
7. If 3 balls are randomly drawn from a bowl containing 6 white and 5 black balls, what
is the probability that one of the balls is white and the other two black?
8. A study is to be conducted in a hospital to determine the attitudes of nurses towards
various administrative procedures. A sample of 10 nurses is to be selected from a total of
the 90 nurses employed by the hospital.
a) How many different samples of 10 nurses can be selected?
b) 20 of the 90 nurses are male and the rest are female. If 10 nurses are randomly selected
from those employed at the hospital, what is the probability that the sample of ten will
include exactly 4 male nurses?
9. A factory produces six complex electronic systems. Two of the six are to be randomly
selected for thorough testing and then classified as defective or not defective.
Find the probability that at least one of the two systems tested will be defective, and the
probability that both are defective
a) if two of the six systems are actually defective, and
b) if four of the six systems are actually defective.
10. (This is a famous problem called the Monty Hall problem from the 1970s which
puzzled many people when posed first, it is based on the American television game
show Let's Make a Deal and named after its original host, Monty Hall.)
Suppose you're on a game show, and you're given the choice of three doors: Behind one
door is a Mercedes (or any dream prize of your choice if you don’t like cars!), and behind
the other two doors are goats. You pick one door, and the host (who knows what's behind
all the doors) opens another door, which has a goat. He then says to you, "Do you want to
switch your choice?" Is it to your advantage to switch your choice?
Here are a series of steps to help you solve the problem using the sample space method.
Suppose the doors are identified by the objects behind them, so the doors can be labelled
M and G1 and G2. The sample space is thus S={M, G1, G2}.
a) Before you pick a door in the beginning, what is the probability that you choose M?
b) Now suppose that you have made a choice, and Mr. Hall has not opened any door yet.
i) If you choose to stay with the first choice, you will win the car if and only if you
had initially chosen M. If you stay with your first choice, what is the
probability that you win the Mercedes (use part (a))?
ii) If Mr. Hall shows you one of the goats and you then switch to the other unopened
door, what will you find behind it if you had initially chosen M?
iii) Answer the question in part (ii) if you had initially chosen one of the goats.
iv) If you switch from your initial choice after being shown the goat, what is the
probability that you win the prize? (Use iii, iv, and think: how many ways?)
v) Which strategy maximizes your probability of winning the Mercedes – stay with
the initial choice or switch to the other door?