代写Compass留学生、代做C/C++编程语言、代写optimization-code、代做C/C++

日期：2018-10-30 10:02

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

1

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.

2

References

[1] V. Addona, S. Wagon, and H. Wilf. How to lose as little as possible. ARS

MATHEMATICA CONTEMPORANEA, 4:29–62, 2011.