Interactive Graphics from Risk and Reward:
The Science of Casino Blackjack

Second Edition

Welcome to the interactive versions of figures and tables from my book, "Risk and Reward: The Science of Casino Blackjack Second Edition".

  1. Download and install Wolfram Player, the software platform enabling the interactivity, by clicking here.
  2. Download any of the interactive items, as listed below, by clicking on its thumbnail. Alternatively, all of them can be downloaded at once, in a Zip file, by clicking here; then unZip that download to retrieve each item in turn.
  3. View and interact with any downloaded item by opening it in the Wolfram Player. Instructions specific to that item are given in its Caption. Should you need help with these procedures, email me at rwerthamer@gmail.com, supplying your contact phone number.

Additional material relevant to the book is available on my website. I created these interactive graphics using the WolframLanguage, they were then tailored expressly for presentation here on the WolframCloud. I am grateful to Wolfram Research and the staff of its Wolfram Media division for their extensive support of this effort, which showcases so impressively their capabilities in interactive publishing.

N. Richard Werthamer

Figure 4.2: Risk of ruin (W, curve with a minimum, blue) and effective yield ratio (Y/B_, curve with a maximum, red) vs. ramp steepness (s); no DAS, no resplits. Plotted for bet spread = 10, capital = 1,000 base bets, rounds = 1,000,000, 4 decks, penetration 0.8. After a computation that regenerates the initial figure, adjust the parameters as desired and then click the small U button in the upper right to plot the revised figure.
Figure 4.2: Risk of ruin (W, curve with a minimum, blue) and effective yield ratio (Y/B_, curve with a maximum, red) vs. ramp steepness (s); no DAS, no resplits. Plotted for bet spread = 10, capital = 1,000 base bets, rounds = 1,000,000, 4 decks, penetration 0.8. After a computation that regenerates the initial figure, adjust the parameters as desired and then click the small U button in the upper right to plot the revised figure.
Figure 4.3: Effective yield vs. risk of ruin, with points MM (solid squares) and HJY (open circles): upper curve for 1,000,000 rounds; lower curve  for 1,000 rounds. After a computation that regenerates the initial figure, adjust the parameters as desired and then click the small U button in the upper right to plot the revised figure.
Figure 4.3: Effective yield vs. risk of ruin, with points MM (solid squares) and HJY (open circles): upper curve for 1,000,000 rounds; lower curve  for 1,000 rounds. After a computation that regenerates the initial figure, adjust the parameters as desired and then click the small U button in the upper right to plot the revised figure.
Figure 6.1: Distribution of outcomes: initial plot shows results after 1000 rounds.  To see how this changes with number of rounds, adjust the slider to select the desired number.  For each plot, the left-most (red) curve represents the non-counter, the right-most (blue) the counter.
Figure 6.1: Distribution of outcomes: initial plot shows results after 1000 rounds.  To see how this changes with number of rounds, adjust the slider to select the desired number.  For each plot, the left-most (red) curve represents the non-counter, the right-most (blue) the counter.
Figure 9.1: Distribution of capital, scaled to its initial value, vs. number of hands played. Shown is the plot for initial capital of 20 unit bets and expected return of 0.01.  Select appropriate settings by clicking on them, then click the U button in the upper right to update the figure. For small numbers of hands the distribution is sharply peaked at a capital ratio of unity; as the number of hands increases the distribution spreads out.  But because of the possibility of ruin, the distribution is always constrained to zero at zero capital. Also shown are plots of the ruin probability (red) and of the expected capital ratio (green), each vs. number of hands.  The rising probability of ruin exactly compensates the decreasing area under the capital ratio surface, i.e., the probability of survival.
Figure 9.1: Distribution of capital, scaled to its initial value, vs. number of hands played. Shown is the plot for initial capital of 20 unit bets and expected return of 0.01.  Select appropriate settings by clicking on them, then click the U button in the upper right to update the figure. For small numbers of hands the distribution is sharply peaked at a capital ratio of unity; as the number of hands increases the distribution spreads out.  But because of the possibility of ruin, the distribution is always constrained to zero at zero capital. Also shown are plots of the ruin probability (red) and of the expected capital ratio (green), each vs. number of hands.  The rising probability of ruin exactly compensates the decreasing area under the capital ratio surface, i.e., the probability of survival.
Figure 9.2:  Distribution of return, scaled to its magnitude at zero count, for various depths.  The plots show that as F increases (blue to red to yellow to green), the distribution widens and flattens. To modify the depth and number of decks interactively, click the Toggle button and select.  At small depths the distributions are narrowly peaked around -1, while as the depth grows they spread further out and have increasing weight at positive returns; the spreading is more pronounced for one deck than for larger numbers.
Figure 9.2:  Distribution of return, scaled to its magnitude at zero count, for various depths.  The plots show that as F increases (blue to red to yellow to green), the distribution widens and flattens. To modify the depth and number of decks interactively, click the Toggle button and select.  At small depths the distributions are narrowly peaked around -1, while as the depth grows they spread further out and have increasing weight at positive returns; the spreading is more pronounced for one deck than for larger numbers.
Fig. 9.3: Survival probability (upper) and yield reduction factor (lower), on a log-log scale, for various number of rounds N and parameter q (ratio of the return distributions drift to its spreading).
Fig. 9.3: Survival probability (upper) and yield reduction factor (lower), on a log-log scale, for various number of rounds N and parameter q (ratio of the return distributions drift to its spreading).
Figure 9.4: Representation of the distribution of capital, for various values of the number of rounds N, labeled when mouse-overed. The plots show that as N increases (blue to red to yellow to green) the distribution shifts, widens and flattens-its mean and variance both increase. To interact, simply move the sliders.
Figure 9.4: Representation of the distribution of capital, for various values of the number of rounds N, labeled when mouse-overed. The plots show that as N increases (blue to red to yellow to green) the distribution shifts, widens and flattens-its mean and variance both increase. To interact, simply move the sliders.
Figure 9.5: Distributions of expected returns: at depth 0.4 (Gaussian curve, blue) and averaged over depths to a penetration 0.8 (peaked curve, red), for representative game parameters.  To select the number of decks, just click on the slider and move it.
Figure 9.5: Distributions of expected returns: at depth 0.4 (Gaussian curve, blue) and averaged over depths to a penetration 0.8 (peaked curve, red), for representative game parameters.  To select the number of decks, just click on the slider and move it.
Figure 9.10: Distributions of true counts at entry and subsequently: entry at shuffle (zero count) is the blue curve, entry at true counts 1 (red), and 2 (yellow); 6 decks.  To interactively choose entry count and number of decks, use the slider.
Figure 9.10: Distributions of true counts at entry and subsequently: entry at shuffle (zero count) is the blue curve, entry at true counts 1 (red), and 2 (yellow); 6 decks.  To interactively choose entry count and number of decks, use the slider.
Figure 9.11: Distribution of true counts, with entry and exit.  The blue curve is without any entry or exit; red with just entry at a count of +1; and yellow for both entry (at +1) and exit (at -1), all for 6 decks.  Note how entry and exit remove most of the weight for negative counts.  To select other entry and exit counts, just move the sliders appropriately.
Figure 9.11: Distribution of true counts, with entry and exit.  The blue curve is without any entry or exit; red with just entry at a count of +1; and yellow for both entry (at +1) and exit (at -1), all for 6 decks.  Note how entry and exit remove most of the weight for negative counts.  To select other entry and exit counts, just move the sliders appropriately.