p1_martingale_report-1

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Dec 6, 2023

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CS 7646 Spring2023 Project 1: Martingale Question 1 : In Experiment 1, based on the experiment results calculate and provide the estimated probability of winning $80 within 1000 sequential bets. Thoroughly explain your reasoning for the answer using the experiment output. Your explanation should NOT be based on estimates from visually inspecting your plots, but from analyzing any output from your simulation. The probability of winning $80 within 1000 successive bets is 100%. The strategy allows us to double our bet amount in the next bet, and the odd of won is 18/38, or 47.36% which is close to 50% winning odd, and each bet is individually independent in odd, so unlimited bet amount will guarantee we reach the $80 winning goal within 1000 successive bets. The mean and median of winning amount both reach to $80 before 1000 simulation successive bets as we calculated in Figure 2 and Figure 3, it indicates that average winning amount of betting converge to $80 before 1000 bets, so the winning amount achieve the goal within our expectation, therefore the stability in the Monte Carlo simulation further validates the result. Question 2 : In Experiment 1, what is the estimated expected value of winnings after 1000 sequential bets? Thoroughly explain your reasoning for the answer. The expected value of winnings after 1000 sequential bets is $80. The mean of Monte Carlo simulation converges to $80 before 1000 sequential bets, so the expected winning amount is $80. The discussion in Question1 validates that the probability of winning $80 in 1000 successive bets is 100%, so the result guarantees the mean of the event is $80. 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? Thoroughly explain why it does or does not. The standard deviation upper and lower lines do not stabilize in the first 200 sequential bets in
my simulation, then they converge to $80. The variation on bets amount in every bank roll across simulations result in un-stabilization of standard deviation in the first 200 runs, but winning amount standard deviation upper and lower lines gradually stabilize and converge to 80 is because the $80 expected winning amount target is reached within 1000 successive bets in every simulation. Question 4 : In Experiment 2, based on the experiment results calculate and provide the estimated probability of winning $80 within 1000 sequential bets. Thoroughly explain your reasoning for the answer using the experiment output. Your explanation should NOT be based on estimates from visually inspecting your plots, but from analyzing any output from your simulation. The mean of the bets in experiment 2 closes to -41.5 according to my simulation result. The experiment 2 scenario fits in the binomial distribution because the winning amount either falls to -$256 or goes up to $80, so assuming probability of winning amount equals to p, thus we have: 80*p+(-256)*(1-p)=-41.5, so the probability of winning $80 within 1000 sequential bets is 0.638. Question 5 : In Experiment 2, what is the estimated expected value of winnings after 1000 sequential bets? Thoroughly explain your reasoning for the answer. The estimated expected value of winnings is the mean in the simulation, so it is equal to -41.5, which can be calculated by binomial distribution mean, 80*0.638+(-256)*0.362=-41.5 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? Thoroughly explain why it does or does not. The upper standard deviation line spikes as the number of bets increases, then the line stabilizes and converges to certain level, the lower standard deviation line declines as the number of bets increase, then the line stabilizes. Both lines reach to max and min, then stable separately. The
bank roll limit 256 and winning amount limit 80 results in the stability of upper and lower line as the number of bets increases, so it guarantees the stability of upper bound line and lower bound line in 1000 simulation of 1000 sequential bets in an episode. 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? One specific random episode cannot be generalized to all simulation results, so we use expected values in experiment to understand the average winning amount over time in martingale simulations, the expected values or mean can easily help us to summarize the performance of different experiments. The expected values can represent the tendency in multiple rounds of simulations in an experiment.
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