Ask The Wizard #226

After analyzing a game played with six decks, I discovered that a player 21 pushing against a dealer blackjack results in a reduction of the house edge by 0.37%. Additionally, a tie in blackjack that pays 3 to 2 decreases the house edge by 0.32%. This means you don't need to alter your playing strategy.

I hope my query finds you well. My understanding of basketball rules and tactics is rather limited, so I sought advice from some friends who are more knowledgeable, but I received conflicting responses. In fact, some answers were completely contradictory. Two theories that emerged from my discussions are (1) the overall field goal percentage in the NBA is closer to 50% ( source ), and (2) there's a possibility that in going for a two-point shot, the shooter may also get fouled and still make the shot. I apologize for not being able to provide a more conclusive answer.

If you're suggesting that the casino alters the game's odds while players are actively using it, I would argue that this is nothing more than a myth. For a slot machine's odds to be changed, the manufacturer would need to physically alter the internal EPROM chip. In the case of server-based games, where adjustments can be made remotely, regulations stipulate that the game must be inactive for a specific duration before any modifications are allowed.

If you are insinuating that a casino might temporarily offer a slot machine with favorable odds to attract new players, only to later replace the EPROM with one that has tighter odds, I would contest that as well. While such a practice could be legally executed, I find it hard to believe they would engage in it. In my survey of slot machines, I observed that casinos generally maintain a consistent level of looseness or tightness in their slots.

This analysis overlooks the fact that these individual scorelines should display a slight negative correlation, and the average points each team concedes is just as crucial as the average points they score. Assuming that 1.5 and 1.2 represent the expected scores in a match, factoring in both offense and defense while disregarding the correlation, we can form a reasonable estimate concerning the probabilities of the three outcomes. Numerous Super Bowl prop bets exist along these lines, such as who will score more touchdowns, field goals, interceptions, etc.

The initial task is to utilize the Poisson distribution to approximate the likelihood of each team scoring a certain number of goals. The general formula states that for a team scoring g goals, with an average of m goals, the probability is e^-m× m^g/g!. In Excel, you can implement the formula poisson(g,m,0). The subsequent table illustrates the probabilities from 0 to 10 goals for both teams, using this formula.

The following step is somewhat tedious, but you’ll need to create a matrix of all 81 possible combinations of scores totaling from 0 to 8 for both teams. This is accomplished by multiplying the likelihood of team A achieving x scores and team B achieving y scores, based on the aforementioned table. The next table presents the probability of every score combination from 0-0 to 8-8.

The following table displays the winner corresponding to each combination of goals, with T symbolizing a tie.

Ultimately, you can leverage the sumif function in Excel to aggregate the appropriate cells for all three potential bet outcomes. In this scenario, the resulting probabilities are:

basketball-reference.com
Nick K. from Scarsdale, NY
source

Les from Fallbrook, CA EPROM Stanford Wong outlines the win/loss/tie probabilities for bets of this nature. For this particular situation, he indicates probabilities of 44%, 30%, and 25%. If anyone is aware of a straightforward formula applicable to this type of problem, I would greatly appreciate it.

Follow-Up: I received an email from Bob P., who always challenges me with math-related inquiries. Here’s what he wrote.

I investigated the distribution concerning the difference between two independent Poisson processes. It’s a 1) A will score more than B 2) B will score more than A

In any case, the question can then be framed as P(Z=0), P(Z>0), and P(Z<0), where Z represents a Skellam distribution with parameters of 1.5 and 1.2.

If you haven’t done so already, you might be pleased to discover that

3) Game finishes as a tie.

Dimitar from Sophia, Bulgaria

-m

× m

The Wikipedia entry for a Skellam touched upon g , which is a complex topic within calculus that quite intimidates me. Therefore, I'll take Bob's word on this one.

The answer and detailed resolution can be accessed on my companion website, Team A Team B