Probability and Statistics
Assignment on
Conditional Probability, CDF, PMF, PDF
1. Show that the probability that exactly one of the events A and B occurs is P (A) + P (B) −
2P (AB).
2. Two players X and Y alternatively throw a pair of die. Rules of the game is as follows:
player X wins if he/she throws a sum of 6 points before Y throws a sum of 7 points and
player Y wins if he/she throws a sum of 7 points before X throws a sum of 6 points. If
player X begins the game, then find the probability that X wins.
P (B c )
3. Prove that P (B|A) ≥ 1 − P (A)
.
4. Each of n urns contains 4 white and 6 black balls, while another urn contains 5 white and 5
black balls. An urn is chosen at random from the (n + 1) urns and two balls are drawn from
it, both being black. The probability that 5 white and 3 black balls remain in the chosen
urn is 1/7. Find n.
5. The following data were recorded from a population: (a) the probability of a male smoker
is 2/5, (b) the probability of selecting a male or a smoker is 7/10, (c) the probability of
selecting a male; if a smoker is already selected is 2/3. Find the probability of selecting (i)
male (ii) a non-smoker (iii) a smoker; if a male is first selected.
6. Consider events A and B such that P (A) = 14 , P (B|A) = 21 , P (A|B) = 14 . Find P (Ac |B c )
and P (A|B) + P (A|B c ).
7. A factory has three machines that manufacture widgets. The percentage of a total days
production manufactured by the machines are 10%, 35% and 55%, respectively. Furthermore
it is known that 5%, 3% and 1% of the outputs of the respective three machines are defective.
What is the probability that a randomly selected widgets at the end of the day’s production
runs will be defective?
8. A jar contains two white balls and three black balls. The ball are drawn from the jar one
by one and placed on the table in the order drawn. What is the probability that they are
drawn in order white, black, black, white, black? [Hint: Use general multiplication rule]
9. A test is used to diagnose a rare disease. The test is only 95% accurate meaning that, in
a person who has the disease it arrive report ‘positive’ with probability 95% and (negative
otherwise) and in a person who does not have a disease, it will report ‘negative’ with pro-
bability 95% (and positive otherwise). Suppose that 1 in 10, 000 children get the disease. A
child is to be tested and given that the test comes to be positive. What is the probability
that the child has disease?
10. There is a box which contains 5 black and 5 white balls. From that box, 5 balls are transferred
at random into an empty second box from which one ball drawn and it is found to be white.
What is the probability that balls transferred from the first box are white?
11. Determine the constant A in the following functions, so that those functions are pmfs/pdfs.
Find CDF (F { X (x)) in each case.
( 2 )x−1
A 3 , if x = 1, 2, 3, ...
(i)fX (x) =
0, otherwise
{ (7) ( 1 )x ( 2 )7−x
A x 3 , if x = 0, 1, 2, ..., 7
(ii) fX (x) = 3
0, otherwise
{ 2
A −(logx)
(iii)fX (x) = x
e 2 , if x > 0
0, otherwise
{ −1
Ax 2 , if 0 < x < 1
(iv)fX (x) =
0, otherwise
−|x|
(v) fX (x) = Ae ; x∈R
12. Balls are drawn from an urn containing 2 white and 5 black balls until a white ball appears.
Let X be the rv that denotes the number of black balls drawn before a white ball appears.
What is the domain and range of X? Give a table of the probability mass function of X.
13. Check if the following functions define CDFs :
(a) FX (x) = 0, if x < 0, = x, if 0 ≤ x ≤ 1/2, and = 1, if x > 1/2.
(b) FX (x) = (1/π) tan−1 x, − ∞ < x < ∞.
(c) FX (x) = 1 − e−x , if x ≥ 0, and = 0, if x > 0.
0, if x ≤ 1
14. For a certain rv X, CDF is defined as FX (x) = K(x − 1) if 1 < x ≤ 3
4
1, if x > 3.
(a) Find the value of K.
(3 )
(b) Find pdf fX (x) and P r 2
≤x≤ 9
2
.
{
α2 x2
15. The pdf for a continuous ‘Rayleigh’ rv X is given by fX (x) = α2 xe− 2 , if x > 0
0, otherwise.
Find CDF of X.
16. The distance
{ covered by a person is assumed to be a continuous rv with pdf
6x(1 − x), if 0 ≤ x ≤ 1 ( )
fX (x) = Find CDF of X. Compute P X ≤ 21 | 13 ≤ X ≤ 32 .
0, otherwise.
Determine k such that P (X > k) = P (X < k).
17. Two unbiased dice are thrown and the rv X denote the sum of faces turned. Construct the
table for pmf and find CDF.
18. The length of time (in minutes) that a certain
{ person speaks over the telephone is found to
− x7
Ke , if x > 0
be a random phenomenon with pdf fX (x) =
0, if x ≤ 0
(a) Find the constant K.
(b) Show that the telephone conversation will last more than m + n minutes given that it
has lasted for at least ‘m’ minutes is equal to the unconditional probability that it will
last more than ‘n’ minutes.
19. Suppose a certain type of small data processing firm is so specialized that some have difficulty
making a profit in their first year operation.
{ The pdf that characterizes the proportion Y
Cy 4 (1 − y)3 , if 0 ≤ y ≤ 1
that make a profit is given by fX (x) =
0, otherwise
After finding the value of C, compute the probability that at most 50% of the firms make a
profit in the first year. Also find the probability that at least 80% of the firms make a profit
in the first year.
20. A bag contains two fair coins and a third coin which is biased. The probability of tossing a
head on this third coin is 3/4. A coin is pulled at random and tossed three times. Let X
be the rv that counts the number of heads obtained in these three tosses. Give the pmf and
CDF.
