78 lines
2.9 KiB
Markdown
78 lines
2.9 KiB
Markdown
# Report
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## Question 1
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In Experiment 1, estimate the probability of winning $80 within 1000 sequential
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bets. Explain your reasoning thoroughly.
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The betting strategy in experiment 1 generates $1 for every win and effectively
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does not create a loss for an incorrect bet. That means we have to lose 921
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times to win less than $80. The odds for that are (20/38)^921, or one could also
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say non-existent.
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## Question 2
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In Experiment 1, what is the estimated expected value of our winnings after 1000
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sequential bets? Explain your reasoning thoroughly.
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Based on the previous answer's reasoning, we have an expected value of $1 times
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18/38 or $0.47 per bet. That means the expected value is $470 for 1000
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consecutive bets. Since the limit is $80, all graphs end at that value. If we
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increase the limit to $1000 and change the graph's dimensions, we can see that
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$470 is the area where the winnings graphs end.
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## Question 3
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In Experiment 1, does the standard deviation reach a maximum value then
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stabilize or converge as the number of sequential bets increases? Explain why it
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does (or does not) thoroughly.
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The more sequential bets reach their final value of $80, the more the standard
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deviation stabilizes and eventually goes to zero. The more bets are ongoing, the
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higher the chances that one of them goes on a losing streak, which results in a
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significant standard deviation. Therefore, the maximum standard deviation is
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likely to occur between the first couple of bets (so that a losing streak can
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build up), but before many bets reach $80.
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## Question 4
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In Experiment 2, estimate the probability of winning $80 within 1000 sequential
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bets. Explain your reasoning using the experiment thoroughly. (not based on
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plots)
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We start at $256. After losing eight times (nine times if we have already built
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some bankroll), we go bankrupt. Consequently, we are at 0.59% ((20 / 38) ** 8)
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risk to lose everything for each dollar we want to earn. Therefore, the
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probability of winning $80 is 0.9941^80 or 62.34%.
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## Question 5
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In Experiment 2, what is the estimated expected value of our winnings after 1000
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sequential bets? Explain your reasoning thoroughly. (not based on plots)
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According to the previous question's answer, we have a 62.34% chance to win $80,
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which leaves us with 27.66% to lose $256. Accordingly, the expected value is
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0.6234 * $80 - 0.3766 * $256 = -$46.53. This result seems to match our
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experiment. After 300 bets, we are on average at -$40, and when we extend the
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timescale to 1000 bets, the graph converges towards $45.
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## Question 6
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In Experiment 2, does the standard deviation reach a maximum value then
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stabilize or converge as the number of sequential bets increases? Explain why it
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does (or does not) thoroughly.
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The standard deviation reaches a maximum value and stabilizes once all runs have
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either bankrupted or reached the $80 goal.
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