Faculty of Basic Sciences
Department of Mathematics
Probability and Statistics exercises
1. In a school, 48% of the students take a foreign language class and 19% of students
take both foreign language and technology. What is the probability that a
student
takes technology given that the students takes foreign language?
2.
The test contains 10 questions, each one with available four different answers,
among which just one is correct. To pass the test at least 5
questions must be
answered correctly. What is the probability that completely unprepared student
will pass the test ?
3.
In the class of 30 students, seven of them don't have done the homework. The
teacher choosed randomly 6 students. What is the chance that at least four of them
have done their homework ?
4.
Three shooters shoot at the same target, each of them shoots just once. The first
one hits the target with a probability of 70%, the second
one with a probability
of 80% and the third one with a probability of 90%. What is the probability that
the shooters will hit the target
a) at least once
b) at least twice ?
5. Based on incidence rate, the following table presents the corresponding numbers
per 100,000 people.
Symptom
Cancer
Total
No
Yes
No
99989 0
99989
Yes
10
1
11
Total
99999 1
100000
Which can then be used to calculate the probability of having cancer when you have
the symptoms:
6. A factory produces an item using three machines—A, B, and C—which account
for 20%, 30%, and 50%
of its output, respectively. Of the items produced by
machine A, 5% are defective; similarly, 3% of machine B's items and 1% of
machine C's are defective. If a randomly selected item is defective, what is the
probability it was produced by machine C?
7.
X is a discrete random variable. The table below defines a probability distribution
for X
X
0
1
2
3
P
0.17
0.14
0.36
0.33
What is the expected value of
X? What is the variance value of X?
8. The random variable
X is given by the following PDF. Check that this is a valid
PDF
and calculate the expected, the variance, the standard deviation values of
X.
9.
Let X be a continuous random variable with the following
𝑓(𝑥) = {
𝑐𝑒
−𝑥
𝑖𝑓 𝑥 ≥ 0
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
where c is a positive constant.
a. Find c.
b. Find the cumulative distribution function of X.
c. Find P(1
