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Ask The Wizard #233
As a tabletop gaming enthusiast, I recently had a conversation with some friends about dice that are not shaped like cubes, specifically non-cubical platonic solids, including d4, d8, d12, and d20. They asserted that these dice would be the only ones that could be conclusively deemed fair. In contrast, I contended that producing them to meet fair standards would prove to be highly challenging. Additionally, the only games suitable for these would be variants of craps that become overly complicated due to the introduction of numerous additional outcomes. Has there ever been a casino game that utilized dice other than traditional six-sided ones?
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If you restrict yourself to regular polygons and seek equal probability for every face, you find yourself limited to the platonic solids. However, by relaxing the requirement to regular polygons, you open up the possibility of incorporating 13. Catalan solids as well.
In response to your earlier question, I can say that I have never witnessed a game in a casino using any dice except for the standard cube. Roughly ten years ago, I observed a game at a gaming exhibition in Atlantic City that seemingly utilized one of the Catalan solids, yet it never made it into an actual casino. Additionally, there’s a game showcased annually at the Global Gaming Expo which employs a spinning top reminiscent of a dreidel, but unfortunately, I’ve never encountered that in a casino setting either. Rhombic triacontahedron If I roll three six-sided dice, what are the chances of getting a straight and what about the chances of landing three of a kind?
There are 216 different combinations to roll three dice. Among those combinations, six will result in a three of a kind (from 1-1-1 to 6-6-6). This gives a probability of 6 out of 216, simplifying to 1 out of 36 for achieving three of a kind. For a straight, there are four potential sets (1-2-3 through 4-5-6). Additionally, there are 3! or 6 ways to arrange the three dice in a straight line. Therefore, the total number of straight combinations is 4 multiplied by 6, equating to 24. Hence, the probability of rolling a straight is 24 out of 216, which simplifies to 1 out of 9.
There are 63In just two weeks, my son managed to score two holes-in-one! What are the statistical chances of that happening? He carries a handicap of 1, with the first hole measuring 151 yards and the second one at 137 yards, on different courses.
John from Pointe Claire, Quebec, Canada
According to Life: the Odds (and How to Improve Them) A 1 handicap is quite impressive, so I won’t significantly reduce the probability compared to PGA Tour professionals. Let's assume your son's odds per par 3 hole is around 1 in 3,000. Typically, a golf course features about four par 3 holes. If your son plays daily, that accounts for 28 par 3 holes each week. The odds of him getting exactly two holes in one would be calculated as follows.
Last week, while I was in Las Vegas engaged in a game of Casino War, I found myself the sole player at the table, with my girlfriend standing behind me observing. I was contemplating my betting strategy when the dealer inadvertently began dealing my cards without me having placed a bet. He quickly retracted the card after realizing the error but without discarding it. I caught a glimpse of the card, which was a Jack, yet I’m unsure if he noticed I saw it. I felt a bit disoriented for a moment, anticipating him to discard the card, and I hesitated out of concern for making a large bet only to attract unwanted attention, ultimately deciding to stick with the minimum bet of $10. To my surprise, I won with the Jack against a lower card of the dealer. In circumstances like these, while a Jack might not guarantee a win, would it have been legal or ethically acceptable to place a larger bet and claim the winnings? I do wish I had gone for a higher stake since the odds seemed favorable, but I feared potential repercussions from a manager or casino security if I ended up winning a significant amount (although we were not being closely monitored). How would you have reacted or what would you suggest doing in a situation like that? combination (28,2)×(1/3000)2×(2999/3000)26= 1 in 24,017.
Although you didn’t explicitly ask for my opinion, I would like to mention that there is a 43.4% edge when your first card is a Jack. The responsibility for this oversight lies with the dealer for inadvertently revealing the card. Contrary to what some casino security staff may incorrectly think, you are completely within your rights to utilize any information presented to you during normal gameplay.
On a moral level, it’s essential to adhere to your own ethical beliefs. You must navigate your own life. Nevertheless, I believe that most players, myself included, would feel justified in raising the stakes in such a scenario. For instance, it’s not up to the players to enforce game security. Additionally, casinos often exploit or depend on players making mistakes. Take the big 6/8 bet in craps as an example; the casinos gladly accept such bets, even when a place bet on 6 or 8 offers the same payout but with significantly better odds. Consider how one would be treated if they accidentally fouled their hand in pai gow poker, even when the proper action is glaringly obvious.
Should you find yourself in a similar situation again, it's wise to exercise caution and remain composed. A sudden jump from a $10 bet to a $500 one could raise numerous alarms. A proficient dealer would likely catch on to the situation, potentially leading to the bet being rejected, or a card being burned.
I noticed a promotion that adds $250 to every taxable jackpot. There’s a double-up feature on the machines, and I intend to double each full house or better until I either lose or surpass $1200. Can you help me calculate the expected value for this game? Thank you.
I am playing 8-5 triple bonus plus Great discovery! Although you didn’t specify the denomination you’re playing, which is a crucial detail, I will assume it's dollars. For the maximum bet of five coins, the number of doubles needed to achieve a win of w (where w < 1200) can be calculated as 1 + int(log(1200) - log(w)) / log(2).
The subsequent table displays the initial hand's pre-double win, pre-double probability, the number of doubles needed, post-double win, and the probability of achieving this post-double win, including the additional $250 bonus. The bottom right cell indicates a return of 115.5%. On average, you will land a jackpot every 297 hands, with an average jackpot value of $1,717.46.
8-5 Triple Bonus Return Table featuring a $250 Bonus for Wins of $1,200 or More
Accurate mathematical strategies and insights for casino games like blackjack, craps, roulette, and numerous others available for play.
| Pre-Double Win | Pays | Pre-Double Probability | Doubles Required | Post-Double Win | Post-Double Probability | Return |
| Royal flush | $4000 | 0.000026 | 0 | $4250 | 0.000026 | 0.02193 |
| Straight flush | $500 | 0.000118 | 2 | $2250 | 0.00003 | 0.013322 |
| 4 aces | $1200 | 0.000235 | 0 | $1450 | 0.000235 | 0.068227 |
| 4 2-4 | $600 | 0.000542 | 1 | $1450 | 0.000271 | 0.078557 |
| 4 5-K | $250 | 0.001629 | 3 | $2250 | 0.000204 | 0.091637 |
| Full house | $40 | 0.010546 | 5 | $1530 | 0.00033 | 0.100842 |
| Flush | $25 | 0.011055 | 6 | $1850 | 0.000173 | 0.063913 |
| Straight | $20 | 0.012738 | 6 | $1530 | 0.000199 | 0.060902 |
| 3 of a kind | $15 | 0.075542 | 7 | $2170 | 0.00059 | 0.256136 |
| Two pair | $5 | 0.123065 | 8 | $1530 | 0.000481 | 0.147101 |
| Jacks or better | $5 | 0.211575 | 8 | $1530 | 0.000826 | 0.252898 |
| Total | 0.447071 | 0 | 0 | 0.003364 | 1.155465 |