ECE 511 Special Examples Summer 2023
SE 2.1: two urns contain, respectively 2 white and 1 red and 2 white and 2 red balls. A person
randomly transfers one ball from the first urn into the second. A ball is now drawn from the
second urn. What is the probability of it being red?
SE 2.2: Three urns contain, respectively, 2 white and 1 red, 2 white and 2 red, and 1 white and 2
red balls. A person randomly transfers one ball from the first urn into the second, then one ball
from the second into the third. A ball is now drawn from the third urn. What is the probability
of it being red?
SE 2.3: In a die experiment, determine the conditional probability of the event {f2} given that
the event even occurred
SE 2.4: A box contains three white balls w1, w2, and w3 and two red balls r1 and r2. Two balls
are removed at random. Determine the probability that the first removed ball is white and the
second red?
SE 2.5: A box contains 6 white and 4 black balls. Remove two balls at random without
replacement. What is the probability that the first one is white and the second one is black?
SE 2.6: Mary has two children. One child is a boy. What is the probability that the other child is
a girl?
SE 2.7: Two boxes B1 and B2 contain 100 and 200 light bulbs respectively. The first box (B1) has
15 defective bulbs and the second 5. Suppose a box is selected at random and one bulb is
picked out.
(a) What is the probability that it is defective? (a priori)
(b) Suppose we test the bulb and it is found to be defective. What is the probability that it
came from box 1? P (B1 / D) = ?
SE 2.8: We have four boxes. Box 1 contains 2000 components of which 5% are defective. Box 2
contains 500 components of which 40% are defective. Boxes 3 and 4 contain 1000 each with
10% defective. We select at random one of the boxes and we remove at random a single
component
(a) What is the probability that the selected component is defective?
(b) We examine the selected component and we find it defective. On the basis of the evidence,
we want to determine the probability that it came from box 2.
SE 2.9: Suppose there exists a (fictitious) test for cancer with the following properties. Let
A : event that the test states that tested person has cancer
B : event that person has cancer
A* : event that test states person is free from cancer
B* : event that person is free is free from cancer
It is known that P{A / B} = P{A* / B*} = 0.95 and P{B} = 0.005. Is the test a good test?
SE 2.10: Three switches connect in parallel operate Independently. Each switch remains closed
with probability p.
(a) Find the probability of receiving an input signal at the output.
(b) Find the probability that switch S1 is open given that an input
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(c) Signal is received at the output.
SE 3.1: A box B1 contains 8 white and 6 red balls and box B2 contains 5 white and 15 red balls, A
ball is drawn from each box. What is The probability that the ball from B1 is white and the ball
from B2 is red?
SE 3.2: A fair die is rolled five times. Find the probability p5 (2) that “six” will show twice.
SE 3.3: A pair of dice is rolled n times. (a) Find the probability that “seven” will not show at all
(b) (Pascal) Find the probability of obtaining double six at least once
SE 3.4: There are four balls numbered 1 to 4 in the urn. Determine The number of
distinguishable, ordered samples of size 2 that can be drawn without replacement. Determine
the number of distinguishable unordered sets
SE 3.5: An odd number of people want to play a game that requires two teams made up of
even number of players. To decide who shall be left out to act as umpire, each of N persons
tosses a fair coin with the following stipulations: If there is one person whose outcome (be it
heads or tails) is different from the rest of the group, that person will be the umpire. Assume
that there are 11 players. What is the probability that a player will be “odd-person out”, i.e.,
will be the umpire on the first play?
SE 3.6: 10 independent binary pulses per second arrive at a receiver. The error probability, i.e.,
a zero received as a one or vice versa) is 0.001. What is the probability of at least one
error/second?
SE 3.7: An order of 104 parts is received. The probability that a part is defective equals 0.1.
What is the probability that the total number of defective parts not exceed 1100?
SE 3.8: Two urns contain balls as follows: #1: 3 Red and 4 Black and #2: 2 Red and 1 Black. A
person randomly transfers one ball from the first urn into the second and then one ball from
the second into the first. A ball is now drawn from the first urn. What is the probability of it
being Red?
SE 3.9: Two assembly lines produce balloons. The capacities of the Line #1 and Line #2 are 100
per min. and 50 per min. respectively. The probability of a balloon being defective in line #1 is
2% whereas the corresponding value for line #2 is 5%.
(a) Find the probability that a balloon randomly chosen in the market is defective.
(b) If a randomly chosen balloon in the market is found to be defective, what is the probability
that it came from assembly line #1
SE 4.1: Toss a coin. ? = {H, T} with P(H) = p. Suppose the r.v X is such that X (T) = 0, X (H) = 1.
Find FX (x).
SE 4.2: A fair coin is tossed twice, and let the r.v X represent the number of heads. P(H) = p. Find
FX (x).
SE 4.3: The pdf of a random Variable x is shown in the figure
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(a) Compute the value of A
(b) Find Fx (x) and sketch the PDF
(c) Compute P [ 2 ≤ x < 3]
(d) Compute P [ 2 < x ≤ 3]
(e) Compute FX (3)
SE 4.4: The pdf of a random variable is shown in the figure.
(a) Compute the value of A. (b) sketch the PDF; (c) Compute P[2 ≤ X < 3];
(d) Compute P[2 < X ≤ 3]; (e) Compute FX (3).
SE 4.5: Consider the random variable X with pdf fX (x) given by
(a) Find A and plot fX (x)
(b) Plot PDF FX (x)
(c) Find point b such that P [X > b] = ? P[X ≤ b]
SE 4.6: Let p = P(H) represent the probability of obtaining a head in a toss. For a given coin, a-
priori p can possess any value in the interval (0,1). In the absence of any additional information,
we may assume the a-priori p.d.f fP (p) to be a uniform distribution in that interval. Now
suppose we actually perform an experiment of tossing the coin n times, and k heads are
observed. This is new information. How can we update fP (p) ?
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SE 4.7: Consider the random variable X with pdf fX (x) as shown in figure below.
(a) Find the value of K
(b) Determine the expression for and plot the PDF FX (x)
(c) Find the probability for X being in the interval -1/2 < X < ?
SE 4.8: A random variable X is defined to be exponential function
fX (x) = (1/ λ) exp (-x/ λ) u(x). The variable Y is a piecewise function of X as shown in the figure
below:
(a) Find and plot the pdf fY (y).
(b) Find and plot the PDF FY (y)
SE 5.1: Let X be a uniform r.v. on (0,1), i,e, X ~ U(0,1) and let Y = 2X + 3. Find fY (y).
SE 5.2: Determine FY (y) and fY (y) if Y = aX + b
SE 5.3: (Square Law Detector) . Solve for FY y) and fY (y) given Y = X2
SE 5.4: If fX (x) is U (-1, 1) and Y = X2, determine FY (y) and fY (y)
SE 5.5: Find fY (y) in terms of fX (x) if