ML4T/martingale/martingale.md

# Report

## Question 1

In Experiment 1, estimate the probability of winning $80 within 1000 sequential
bets. Explain your reasoning thoroughly.

The betting strategy in experiment 1 generates $1 for every win and effectively
does not create a loss for an incorrect bet. That means we have to lose 921
times to win less than $80. The odds for that are (20/38)^921, or one could also
say non-existent.

## Question 2

In Experiment 1, what is the estimated expected value of our winnings after 1000
sequential bets? Explain your reasoning thoroughly.

Based on the previous answer's reasoning, we have an expected value of $1 times
18/38 or $0.47 per bet. That means the expected value is $470 for 1000
consecutive bets. Since the limit is $80, all graphs end at that value. If we
increase the limit to $1000 and change the graph's dimensions, we can see that
$470 is the area where the winnings graphs end.

![](figure_1.png)

## Question 3

In Experiment 1, does the standard deviation reach a maximum value then
stabilize or converge as the number of sequential bets increases? Explain why it
does (or does not) thoroughly.

The more sequential bets reach their final value of $80, the more the standard
deviation stabilizes and eventually goes to zero. The more bets are ongoing, the
higher the chances that one of them goes on a losing streak, which results in a
significant standard deviation. Therefore, the maximum standard deviation is
likely to occur between the first couple of bets (so that a losing streak can
build up), but before many bets reach $80.

![](figure_2.png)

![](figure_3.png)

## Question 4

In Experiment 2, estimate the probability of winning $80 within 1000 sequential
bets. Explain your reasoning using the experiment thoroughly. (not based on
plots)

We start at $256. After losing eight times (nine times if we have already built
some bankroll), we go bankrupt. Consequently, we are at 0.59% ((20 / 38) ** 8)
risk to lose everything for each dollar we want to earn. Therefore, the
probability of winning $80 is 0.9941^80 or 62.34%.

## Question 5

In Experiment 2, what is the estimated expected value of our winnings after 1000
sequential bets? Explain your reasoning thoroughly. (not based on plots)

According to the previous question's answer, we have a 62.34% chance to win $80,
which leaves us with 27.66% to lose $256. Accordingly, the expected value is
0.6234 * $80 - 0.3766 * $256 = -$46.53. This result seems to match our
experiment. After 300 bets, we are on average at -$40, and when we extend the
timescale to 1000 bets, the graph converges towards $45.

![](figure_4.png)

## Question 6

In Experiment 2, does the standard deviation reach a maximum value then
stabilize or converge as the number of sequential bets increases? Explain why it
does (or does not) thoroughly.

The standard deviation reaches a maximum value and stabilizes once all runs have
either bankrupted or reached the $80 goal.

![](figure_5.png)