Ask The Wizard #196

Absolutely! As I elaborated in my part regarding the Kelly Criterion , there exists an ideal bet size for every wager where the player has an advantage, aimed at striking the right balance between risk and potential rewards. Betting precisely the Kelly recommended amount promises the most significant growth for a player's bankroll, assuming average luck.

For example, in full pay deuces wild , with a return of 100.76%, the ideal betting amount for each hand is 0.03419% of the total bankroll. Nowadays, if you manage to find full pay deuces wild games, they are likely to be available only at the quarter denomination, but if you have the freedom to bet whatever you like, wagering 0.03419% of your total bankroll would be the most beneficial for long-term growth. For someone with a bankroll of $3,656, a quarter denomination game represents the optimal Kelly bet size.

As I mention in my section about Kelly, the best betting amount is the one that maximizes the expected logarithm of the bankroll after a wager, which I will refer to as the Kelly Utility. Typically, the Kelly Utility reaches its peak when adhering to the optimal strategy. Nonetheless, one notable exception occurs with a combination of five 3s to 9s, accompanied by three deuces. Specifically, let’s examine 22277. The expected value of retaining only the deuces is 15.057354, while holding onto the five of a kind is consistently worth exactly 15.

The following table presents both the traditional expected value and the Kelly Utility when holding onto the three deuces. For any given hand during the draw, the Kelly Utility can be calculated using the formula p*log(1+0.0003419*w), where p symbolizes probability, and w represents the win.

Player Holds Three Deuces

Hand	Pays	Combinations	Probability	Return	Kelly Utility
Four deuces	200	46	0.042553	8.510638	0.001222
Wild royal	25	40	0.037003	0.925069	0.000137
Five of a kind	15	67	0.06198	0.929695	0.000138
Straight flush	9	108	0.099907	0.899167	0.000133
Four of a kind	5	820	0.758557	3.792784	0.000563
Total		1081	1	15.057354	0.002193

The subsequent table displays the same statistics for retaining the five of a kind.

Player Holds Five of a Kind

Hand	Pays	Combinations	Probability	Return	Kelly Utility
Four deuces	200	0	0	0	0
Wild royal	25	0	0	0	0
Five of a kind	15	1	1	15	0.002222
Straight flush	9	0	0	0	0
Four of a kind	5	0	0	0	0
Total		1	1	15	0.002222

You can observe that the Kelly Utility is more advantageous when keeping the established five of a kind, standing at 0.002222 compared to 0.002193. For this specific hand, opting to retain the five of a kind is the preferred choice according to the Kelly Criterion for bankrolls up to 13,290 units or, for those betting quarters, up to $16,613.

As previously stated, the optimal Kelly bet size according to the ideal strategy is 0.03419% of the bankroll. If a player sticks to the optimal strategy but opts to keep a dealt 22233 to 22299, the ideal bet size adjusts to 0.03434% of the bankroll. The bankroll growth anticipated for the player following the optimal strategy stands at 0.0002605% with each wager made. Conversely, a Kelly player can expect 0.0002615% growth per wager. After 40,000 hands, the player adhering to both the optimal strategy and Kelly bet sizing can expect their bankroll to grow by 10.98%. Meanwhile, the conservative player keeping a dealt 22233 to 22299, applying Kelly bet sizing correspondingly, can foresee a growth of 11.03% over the same 40,000 hands.

As such, I argue that in certain instances, it is justifiable to diverge from the optimal strategy in favor of a more cautious approach. I only hope Rob Singer doesn’t catch wind of this.

Before any cards are dealt, the likelihood of not seeing any of three specific ranks on the flop is combin (40,3)/combin(52,3) = 9880/22100 = 44.71%. Hence, this individual had a 10.59% edge.

I inquired with a contact of mine who works at a casino employing this technology. Beyond Treasure Island, this system is also in operation at casinos located in California, Michigan, and Mississippi. Here’s what he shared with me,

"No changes can be executed while there are credits on the machine. The slot machine will refuse to accept any alterations while credits remain active. In Nevada, the machine must also remain idle for four minutes before and after making any changes. Additionally, it isn’t entirely clear to bystanders at the slot machine. A black window appears stating 'Remote Configuration In Progress' (or words to that effect).

We primarily utilize it to adjust the available denominations on our machines. It’s akin to how table games increase minimum bets during busy times; we withdraw lower denominations at the start of the weekend and reinstate them on Monday morning.\"

Rest assured, the slot manager cannot tighten a game against you simply because of personal bias. As long as you have credits in the game, no alterations can be made.

I forwarded this query to Brian, a former gaming regulator and current casino manager. Here’s what he revealed,

The casino would be unaware if a person is undocumented. Provided he possesses a valid passport, his jackpot would be honored. The individual may be unaware of this, scared, or lacking valid identification. Upon winning $1,200 or more, requiring ID for taxation is mandatory. If he can’t present ID, the jackpot will be retained until he appears to claim it. Frequently, winners simply forget their ID, but sometimes issues can arise, like minors participating in gaming activities. If he doesn’t come forward, the funds must revert back to revenue due to the jackpot not being paid, or the governing rules surrounding abandoned property apply. Also, similar to the U.S., most countries levy taxes on worldwide income. In light of this, the U.S. has tax treaties with several nations to notify them of any winnings in the U.S., ensuring Uncle Sam collects his share.

For those unfamiliar, a 'set' refers to achieving three of a kind after the flop, including a pocket pair. The odds of not hitting a set are (48+combin(48,3))/combin(50,3) = 17,344/19600 = 88.49%. This indicates a probability of making a set at 11.51%. Hence, with 2,787 pairs, one would expect to make a set approximately 320.8 times. You are currently falling short of expectations by 47.8 sets. The variance can be calculated using n × p × (1-p), where n is the number of hands, and p is the probability of making a set. In this instance, the variance calculates to 2,787 × 0.1176 × 0.8824 = 283.86. The standard deviation, calculated as the square root of that, amounts to 16.85. Consequently, you are 47.8/16.85 = 2.84 standard deviations below expected outcomes. The probability of experiencing such poor luck or worse can be referenced in any Standard Normal distribution table, or using Excel where norsdist(-2.84) reveals a rarity of about 0.002256, translating to roughly 1 in 443.