Description
1 Part 1: How to Loose as Little as Possible
Skim through the first few pages of reference [1], which is available on Compass.
We have two players Alice and Bob, each have a coin that they can toss. The
probability of heads for Alice’s coin is q, and that for Bob’s coin is p, where
q < p. They agree to to the following game
Each will toss their respective coins n times, Alice wins if and only
if she gets more heads than Bob at the end of the n-many tosses.
Obviously, the game favors Bob over Alice, but reference [1] studies the following
problem
What is the optimal value of n that will maximize Alice’s chance of
winning?
You should take some time to understand equation 1.1 of the paper. You should
also understand what Theorem 2.2 of the paper is trying to say about the shape
of the plot shown in figure 1 in a general setting (i.e. for any value of q < p; not
just q = 0.18 and p = 0.2). This will give you an idea of how you might want
to write the optimization-code for this problem described in subsection 1.1.
1.1 The Optimization Problem of Reference [1]
In this part of the programming assignment you are going to write a C++ program that verifies the results reported in [1]. For this programming assignment
I want you to define an appropriate Class structure and member functions that
picks the optimal value of n for values of p and q that are read at the command
line. A sample output is shown in figure 1. Depending on your implementation,
it might take longer on your computer, be patient!
2 Part 2: The Experimental Verification of Reference [1]
In the previous part of this programming assignment you wrote C++ code that
computed the (combinatorial) expression for the probability of Alice’s winning
the “n-coin-toss” game. In this programming assignment you are going to write
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Figure 1: Sample output. Depending on your implementation, it might take
some time to run on your computer. Be patient.).
a simulation that “experimentally” verifies the (combinatorial) expression used
in the previous programming assignment.
For a given p and q, your simulation should try a range of possible values for
n (say, from 1 to 25), and repeat the “n-coin-toss” game a large number of times
no of trials (say, no of trials = 106
times) – you then count the number of times
Alice wins, and divide it by the number of repetitions (i.e. 106
) and present
that as the probability of Alice’s winning. You should plot this probability as
a function of n, and compare it with the (combinatorial/theoretical) expression
in [1]. I am looking for a plot that looks like what is shown in figure 2. Depending
on your implementation and the number of repetitions of the game, it might
take longer on your computer, be patient!
0 5 10 15 20 25 30
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Number of Coin Tosses in Each Game
Alice’s Probability of Winning
Experiment
Theory
Figure 2: Sample output q = 0.18, p = 0.2, no of trials = 500,000. Depending
on your implementation, it might take some time to run on your computer. Be
patient. This run took a little over 6 minutes on my computer.
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References
[1] V. Addona, S. Wagon, and H. Wilf. How to lose as little as possible. ARS
MATHEMATICA CONTEMPORANEA, 4:29–62, 2011.
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