p1_martingale_report

pdf

School

University Of Arizona *

*We aren’t endorsed by this school

Course

523

Subject

Economics

Date

Feb 20, 2024

Type

pdf

Pages

Uploaded by PrivateIbis3915

Project 1 Martingale Dhvani Mehta dmehta83@gatech.edu Abstract— This project is about performing probabilistic experiments involving an American Roule±e wheel . This report includes the experimental hypothesis, design and findings with supporting graph figures. Question 1 In Experiment 1, based on the experiment results, calculate and provide the estimated probability of winning $80 within 1000 sequential bets. Answer 1 The probability of winning $80 within 1000 sequential bets is almost 1. The Martingale strategy has a winning probability of 9/19 for american roule±e. If the bet amount is $1, the winning amount will be $1 too. That means at every single win $1 will be added and that concludes by winning $80 for 80 successful spins out of 1000. The Figure 1 shows that, around 200 spins it stables at $80 for 10 runs. It doesn’t have the limit on bank roll which means within 1000 sequential bets, winning $80 is very certain in fact the mean and median values in Figure 2 and 3 shows how quickly it se±les to $80 (within 150-200 spins). 1

Figure 1— 10 runs for 1000 simulation spins (showing 300 spins for each run) Question 2 In Experiment 1, what is the estimated expected value of winning after 1000 sequential bets? Answer 2 The expected value of winning after 1000 sequential bets is the mean value $80 according to Figure 2. The mean of the discrete random variable X is also called the expected value of X. Notationally, the expected value of X is denoted by E(X). Formula for calculating Mean: E(X) = Σ [ xi * P(xi) ] where xi is the value of the random variable for outcome i, and P(xi) is the probability that the random variable will be equal to outcome i. (From Stat Trek ) 2

Figure 2— 1000 runs for 1000 simulation spins with mean, (mean + std) and (mean - std) 3

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Figure 3— 1000 runs for 1000 simulation spins with median, (median + std) and (median - std) Question 3 In Experiment 1, do the upper standard deviation line (mean + stdev) and lower standard deviation line (mean – stdev) reach a maximum (or minimum) value and then stabilize? Do the standard deviation lines converge as the number of sequential bets increases? Answer 3 Yes, the upper standard deviation line (mean + stdev) and lower standard deviation line (mean - stdev) reach a maximum value then stabilize. The standard deviation lines converge as the number of sequential bets increases. The reason those values reach maximum is because the number of sequential bets increases, the bet amount increases as well, thus the mean will go high as well. The convergence happens because once the winning amount reaches $80, there won’t be any bets made and the winning amount will be set to $80 for the rest of the spins and deviation will be 0. And also this simulation has unlimited bankrolls and no limits on bet amount. Question 4 In Experiment 2, based on the experiment results calculate and provide the estimated probability of winning $80 within 1000 sequential bets. Answer 4 Based on the experiment results, each round ends with winning $80 or -$256. With the size of 1001X1000 array ( 1 episode = 1000 successive bets), the number of runs that got $80 within a full array is 563283 and not ge±ing $80 is 437717. The probability of winning $80 is 563283/1001000 = .5627 i.e. 56%. 4

Question 5 In Experiment 2, what is the estimated expected value of winnings after 1000 sequential bets? Answer 5 The expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values . (From Investopedia ) EV = ∑ P(X i ) x X i The expected value of winning after 1000 sequential bets based on previously calculated winning probability (answer 4), 56% * $80 + 44% * -$256 = -$67.84. This above expected value is close enough to the mean value of winning $80 for 1000 sequential bets as well. Question 6 In Experiment 2, do the upper standard deviation line (mean + stdev) and lower standard deviation line (mean – stdev) reach a maximum (or minimum) value and then stabilize? Do the standard deviation lines converge as the number of sequential bets increases? Answer 6 Yes, the upper standard deviation line (mean + stdev) and lower standard deviation line (mean - stdev) reach a maximum value then stabilize. However the standard deviation lines don’t converge for experiment 2 because of the bankroll limit and the limit on be±ing amount as well. As the sequential bet increases, the bet amount will increase (it will increase by double if the previous bet was lost). Having a constraint on bankroll of $256, if all the sequential bets are lost, there are no further bets that can be made. So with the two conditions 1) limited bankroll ($256) 2) limit on bet amount, anyone can 5

win the most $80 and then stabilize or win -$256 and then stabilize as shown in figures 4 and 5. Figure 4— 1000 runs for 1000 simulation spins with mean, (mean + std) and (mean - std) with $256 bankroll 6

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Figure 5— 1000 runs for 1000 simulation spins with median, (median + std) and (median - std) with $256 bankroll Question 7 What are some of the benefits of using expected values when conducting experiments instead of simply using the result of one specific random episode? Answer 7 Based on the answers 2 and 5, the expected value is almost the same as the mean value or it can be a weighted average value for a given number of outcomes. For martingale strategy here, if we talk about an experiment where there is unlimited bankroll and no limits on bet amount, the win of $80 is certain by 1000 spins in fact no need to have 1000 spins, it can be achieved within 200 spins. The expected value is very beneficial when someone is planning for a longer term to know what can be their profit/win over a period of time. Sometimes it happens that you never get expected value as a result of a spin. The Martingale strategy with limited bankroll and bet amount can make you lose against the house while playing American roule±e, especially because of the extra 00 block. This analysis can be derived by knowing the expected value for 1000 episodes which is a negative amount. The expected value logic works be±er on a large number of repetitive runs. R EFERENCES 1. American Roule±e Wheel 2. Expected value: Stat Trek , Investopedia 3. Expected value benefits 7

Related Documents

2162++more+FX+exercises_2023.docx

assignment 2.docx

Assignment 1- How your food choices affect the environment..docx

Assignment2 Solutions Chap 2 Forecasting.pdf

Practice exam - chs 4-7 material.pdf

Unit_III_Assignment_Worksheet - Microecon.docx

Journal unit 5.docx

L3 Review SOLUTIONS.xlsx

ECON 111 (07) Midterm A KEY Fall2022.doc

ECON 111 (07) Midterm B KEY Fall2022.doc

4.5 Quiz_ Gross Domestic Product_ Principles of Macroeconomics 32870-933.pdf

7.5 Quiz_ The Keynesian Model_ Principles of Macroeconomics 32870-933.pdf

Recommended textbooks for you

Managerial Economics: A Problem Solving Approach

Economics

ISBN:9781337106665

Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor

Publisher:Cengage Learning

Managerial Economics: Applications, Strategies an...

Economics

ISBN:9781305506381

Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris

Publisher:Cengage Learning

Microeconomic Theory

Economics

ISBN:9781337517942

Author:NICHOLSON

Publisher:Cengage

MACROECONOMICS FOR TODAY

Economics

ISBN:9781337613057

Author:Tucker

Publisher:CENGAGE L

Survey Of Economics

Economics

ISBN:9781337111522

Author:Tucker, Irvin B.

Publisher:Cengage,

Economics For Today

Economics

ISBN:9781337613040

Author:Tucker

Publisher:Cengage Learning

SEE MORE TEXTBOOKS

Recommended textbooks for you

Managerial Economics: A Problem Solving Approach
Economics
ISBN:9781337106665
Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:Cengage Learning
Managerial Economics: Applications, Strategies an...
Economics
ISBN:9781305506381
Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Publisher:Cengage Learning
Microeconomic Theory
Economics
ISBN:9781337517942
Author:NICHOLSON
Publisher:Cengage
MACROECONOMICS FOR TODAY
Economics
ISBN:9781337613057
Author:Tucker
Publisher:CENGAGE L
Survey Of Economics
Economics
ISBN:9781337111522
Author:Tucker, Irvin B.
Publisher:Cengage,
Economics For Today
Economics
ISBN:9781337613040
Author:Tucker
Publisher:Cengage Learning