CHAPTER 1
1. A box contains 6 red pens, 4 blue pens and 5 black pens. Two pens are randomly selected
one after another without replacement.
(a) Draw a tree diagram.
(b) What is the probability that both pens selected are red? answer: 1/7
(c) What is the probability that second pen is blue, given that the first pen was red?
answer: 2/7
2. A teacher has a box of markers consisting of 3 green markers, 6 black markers and 5 red
markers. Two markers are randomly chosen one after another without replacement.
(a) Draw a tree diagram.
(b) What is the probability that both markers are black? answer: 15/91
(c) What is the probability that the second marker is red, given that the first was green?
answer: 5/13
(d) What is the probability that the two markers are of different colors? answer: 7/10
3. A jar contains 2 blue marbles, 3 yellow marbles and 5 red marbles. A marble is drawn at
random, noted for its color, then put back into the jar. This process is repeated a second
time.
(a) Draw a tree diagram.
(b) What is the probability that both marbles drawn are red? answer: 1/4
(c) What is the probability that the first marble is yellow and the second is blue?
answer: 3/50
(d) What is the probability that the two marbles are of the same color? answer: 0.38
4. A game spinner is divided into 4 equal sections labelled as 1, 2, 3, and 4. A player spins the
spinner twice, and each number is equally likely on every spin.
(a) What is the probability that the first spin is a 1 and the second spin is an even number?
answer: 1/8
(b) What is the probability that the first spin is a 3 and the second spins also land on the
number 3? answer: 1/16
5. A system where two servers, Server A and Server B, that handling requests for an online
service. The probability of Server A going down is 0.1 (i.e., 10% chance that Server A fails),
and the probability of Server B going down is 0.2 (i.e., 20% chance that Server B fails). If the
failures of these servers are independent events, calculate:
(a) the probability that both servers will fail at the same time? answer: 0.02
(b) the probability that at least one server will fail? answer: 0.28
6. In a data center, the two hard disks, Disk A and Disk B, which are used to store critical
data. Each hard disk has a probability of failure. The probability that Disk A fails is 0.05
(5%), and the probability that Disk B fails is 0.07 (7%).
(a) Would the failures of Disk A and Disk B be independent? answer: yes
(b) What is the probability that neither disk fails? answer: 0.8835
(c) What is the probability that at least one disk fails? answer: 0.1165
CHAPTER 2
1. Consider the experiment of tossing a coin three times. Let X is the number of heads.
Construct the probability distribution of X.
2. The probability distribution of a discrete random variable X is given by
x
P ( X = x)
0
1
12
1
1
4
2
1
3
3
1
3
Find
(a) P (1 X 3) answer: 2/3
(b) P ( X 2 )
answer: 2/3
(c) E ( X )
answer: 2.08
(d) Var ( X )
answer: 0.243
3. Suppose a charity organization is mailing printed return-address stickers to over one
million homes in Ethiopia. Each recipient is asked to donate either $1, $2, $5, $10,
$15, or $20. Based on past experience, the amount a person donates is believed to
follow the following probability distribution:
x
P ( X = x)
$1
0.1
$2
0.2
$5
0.3
$10
0.2
$15
0.15
$20
0.05
(a) What is the probability that a randomly selected person donates at least $5? answer: 0.7
(a) What is the probability that a person donates at most $5? answer: 0.6
(b) What is the probability that a person donates more than $10? answer: 0.2
(c) What is the expected value (mean) of the donation? answer: 7.25
(d) Find the standard deviation of the donation. answer: 5.44
4. Suppose a mobile app company offers optional in-app purchases to users. Based on
past data, the company found that users spend the following amounts per month
with the following probabilities:
x
P ( X = x)
RM 0
0.3
RM 10
0.2
RM 15
0.25
RM 20
0.15
RM 30
0.08
RM 50
0.02
(a) Define the random variable, X. answer: X = Amount (in RM) a user spends on
in-app purchases per month.
(b) What is the probability that a user spends not more than RM 30 in a month?
answer: 0.98
(c) What is the probability that a user spends greater than RM 15 in a month?
answer: 0.25
(d) What is the expected amount a user spends per month? answer: RM12.15
(e) What is the standard deviation of the amount spent per month? answer: RM10.52
5. A small sports shop tracks the number of soccer balls it sells each day. Based on
past sales records, the owner has created a probability distribution for the number
of soccer balls sold on any given day:
x
P ( X = x)
0
0.05
1
0.15
2
0.25
3
0.30
4
0.15
5
0.10
(a) Define the random variable, X. answer: X = number of soccer balls sold per day
(b) What is the probability that the shop sells at least 3 balls in a day? answer: 0.55
(c) What is the probability that the shop sells between 1 and 4 balls in a day?
answer: 0.85
(d) What is the expected number of soccer balls sold per day? answer: 2.65 balls
(e) Find the standard deviation of the number of soccer balls sold per day. answer: 1.32
6. Let the probability density function of a random variable X be
kx + 1 , −1 x 1
f ( x) =
, otherwise
0
(a) find k so that f(x) is a valid pdf.
(b) P ( 0 x 1)
answer: k=1
answer: 1
(c) Calculate the mean value.
answer: 2/3
7. Let the probability density function of a random variable X be
kx
f ( x) =
0
2
, 0 x2
, otherwise
(a) find k so that f(x) is a valid pdf. answer: k=1/2
(b) P (1 x 2 )
answer: 3/4
(c) Calculate the mean value.
answer: 4/3
8. Let the probability density function of a random variable X be
, 0 x 1
ax
f ( x ) = c ( 2 − x ) , 1 x 2
0
, otherwise
(a) find k and c.
(b) P (1 x 2.5)
(c) Calculate the mean value.
answer: a=1, c=1
answer: 0.5
answer: 1
9. In a certain computer system, the time (in seconds) a user waits for a file to be
retrieved depends on server load. The wait time has the following probability
density function:
, 0 x2
kx
f ( x ) = c ( 4 − x ) , 2 x 4
0
, otherwise
(a) Define the random variable, X. anwer: X = time (in seconds) a file is retrieved
(b) Find the values of k and c so that f(x) is a valid probability density function.
answer: k=1/4, c=1/4
(c) What is the probability that a file is retrieved between 1 and 3 seconds? answer: 0.75
(d) Calculate the mean time a file to be retrieved. answer: 2
10. The number of hours a laptop runs on battery before requiring a recharge. Based on
usage patterns, the battery life follows the piecewise probability density function:
2
, 0 x 1
kx
f ( x ) = c ( 2 − x ) , 1 x 2
0
, otherwise
(a) Define the random variable, X. answer:X=number of hours a laptop runs on full charge.
(b) Find the values of k and c so that f(x) is a valid probability density function. answer: 1
(c) Find the probability that the laptop lasts between 0.5 and 1.5 hours on a full charge.
answer:
(d) Calculate the mean number of hours a laptop runs. answer: 2
(e) Calculate the variance number of hours a laptop runs.
0
