Fix mistake in previous solution and finish report for project 1.

martingale/martingale.md

@@ -1,7 +1,77 @@
-# Report 
+# 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)
+

martingale/martingale.py

@@ -90,7 +90,7 @@ def experiment_1_figure_2(number_runs=1000):
     configure_plot()
     runs = np.array([run_experiment() for _ in range(number_runs)])
     winnings_mean = runs.mean(axis=0)
-    winnings_std = winnings_mean.std()
+    winnings_std = runs.std(axis=0)
 
     plt.plot(winnings_mean, linewidth=0.7)
     plt_std_setting = {'ls': '-', 'color': 'blue', 'linewidth': 0.3}
@@ -103,7 +103,7 @@ def experiment_1_figure_2(number_runs=1000):
 def experiment_1_figure_3(runs, figurename='figure_3.png'):
     configure_plot()
     winnings_median = np.median(a=runs, axis=0)
-    winnings_std = winnings_median.std()
+    winnings_std = runs.std(axis=0)
     plt.plot(winnings_median, linewidth=0.7)
     plt_std_setting = {'ls': '-', 'color': 'blue', 'linewidth': 0.3}
     plt.plot(winnings_median + winnings_std, **plt_std_setting)
@@ -115,7 +115,7 @@ def experiment_2_figure_4(number_runs=1000):
     runs = np.array([run_experiment(initial_credit=256)
                     for _ in range(number_runs)])
     winnings_mean = runs.mean(axis=0)
-    winnings_std = winnings_mean.std()
+    winnings_std = runs.std(axis=0)
     plt.plot(winnings_mean, linewidth=0.7)
     plt_std_setting = {'ls': '-', 'color': 'blue', 'linewidth': 0.3}
     plt.plot(winnings_mean + winnings_std, **plt_std_setting)