Description
1. [10 Marks] Consider the digital binary symmetric communication channel that works as
follows. The transmitter transmits a sequence of binary symbols. For this problem, consider the transmission of just one symbol X ∈ {x0, x1}. x0 is transmitted with probability
p0 = 0.2 and x1 is transmitted with probability p2 = 0.8. The channel can introduce an
error by ‘flipping the symbol’, i.e., x0 will be received as symbol y0 with probaility 0.6
and as y1 with probability 0.4. Similarly, x1 will be received as symbol y1 with probaility
0.6 and as y0 with probability 0.4. This is represented in the Figure. 1
x0 y0
x1 y1
0.6
0.4
0.6
0.4
Figure 1: Binary Symmetric Channel
Find the following
(a) P(y0), P(y1) i.e., probabilities of receiving y0, y1 at receiver
(b) P(x0|y0), P(x0|y1), P(x1|y0), P(x1|y1).
2. [20 Marks] A person possesses five coins, two of which are are heads on both sides, one
has a tail on both sides two are normal fair coins. With eyes shut, the person picks a coin
at random and tosses it.
(a) What is the probability that the lower face (the face that is not seen) of the coin is a
head?
(b) The person opens eyes and sees that the coin is showing heads on top. What is the
probability that the lower face of the coin is a head?
(c) The person shuts the eyes and tosses the same coin again. What is the probability
that the lower face of the coin is a head?
(d) The eyes are opened, and the person sees that the coin is showing heads. What is
the probability that the lower face of the coin is a head?
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3. [15 marks] There are R brown balls and B black balls in an urn. Balls are drawn at
random without replacement. Let Ak be the event that a brown ball is drawn for the first
time on the k-th draw. Find pk, the probability of Ak. Now consider the case when B
and R are increased to ∞ while keeping α = R/(B + R). Find pk as B + R → ∞.
4. [15 marks] There are n urns of which the r-th urn contains r − 1 brown balls and n − r
black balls. You pick an urn at random and pick two balls at random without replacement.
What is the probability that the second ball is black. What is conditional probability that
the second ball is black given that the first ball is black.
5. [20 marks] 10% of the surface area of a sphere is white and the rest is black. There
are no assumptions on how this white part is distributed on the surface. Prove that it
is always possible to inscribe a cube with all its vertices black. Think of a randomly
inscribed cube. Let Ai be the probability a random vertex is white. Now obtain an upper
bound on the probability that at least one of the vertices is white. Show that this strictly
less than one. This proves that there is at least one cube with all black vertices.
6. [40 marks]
(a) You want to estimate the fish population in Powai lake. You do the following: Catch
100 fish, mark them and release them back into the lake. Allow the fish to mix well
and then you catch 100 fish again. Of these 10 are those that were marked before.
Assume that the fish population is n and has not changed between the catches. Find
the probability of the outcome of your experiment.
(b) Now consider a generalisation. Catch m fish, mark them and release them back
into the lake. Allow the fish to mix well and then you catch m fish. Of these p are
those that were marked before. Assume that the actual fish population in the lakes
is n and has not changed between the catches. Let Pm,p(n) be the probability of the
event (for a fixed p recatches out of m) coming from n fish in the lake. Generate a
plot for Pm,p(n) as a function of n for the following values of m and p : m = 100
and p = 10, 20, 50, 75. For each of these p, use the plots to estimate (educated
guess) the actual value of n. Call these four estimates nˆ1, . . . , nˆ4.
(c) Now for each of the four nˆi
, simulate the previous problem. One simulation will
for case i will do the following. Take nˆi fish in the pond, mark m = 100 of these,
mix up the fish and catch m = 100 random fish. Repeat the experiment 500 times
for each i, calculate the sample average from the 500 experiments and compare
with the corresponding p from the previous question. Comment on the comparison
between them.
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