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Ask The Wizard #233
I’m a tabletop gamer, and was having some discussion with my friends about non-cubical platonic solid dice (If you’re a big enough nerd, that means d4, d8, d12, and d20). They argued that those would be the only ones that would be demonstratively fair. I argued that manufacturing them to be fair would be entirely too difficult. Also, the only games would be craps variants rendered overly cumbersome due to the number of extra outcomes. Has any casino ever had a game that used non-traditional six sided dice?
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If you limit yourself to the regular polygons, and want every face to have the same probability, then you are limited to the platonic solids. However, if you can lift the regular polygon requirement, then you can add the 13 Catalan solids as well.
To answer your other question, no, I have never seen a game actually in a casino that used any dice other than cubes. About ten years ago I saw a game demonstrated at a gaming show in Atlantic City that I think used a Rhombic triacontahedron, one of the Catalan solids, but I don’t think it ever made it to a casino floor. There is a game I see year after year at the Global Gaming Expo that uses a spinning top (like a dreidel), but alas, I’ve never seen that in a casino either.
If I roll three six-sided dice, what are the odds of rolling a straight and, also, what are the odds of rolling a three of a kind?
There are 63=216 ways to roll three dice. Six of those combinations will result in a three of a kind (1-1-1 to 6-6-6). So the probability of a three of a kind is 6/216 = 1/36. There are four possible spans for a straight (1-2-3 to 4-5-6). There are also 3!=6 ways to arrange the three dice in a straight. So, there are 4*6=24 straight combinations. Thus the probability of a straight is 24/216 = 1/9.
My son just made two holes in one, in the span of 2 weeks. What are the odds. My son has a 1 handicap. the 1st 151 yards and the 2nd 137 yards, at two different courses.
According to Life: the Odds (and How to Improve Them) by Gregory Baer, the odds of a hole in one on a par 3 hole in the PGA tour is 1 in 2491. I believe those distances fall in the par 3 range.
A 1 handicap is darn good, so I'm not going to give much of a discount compared to PGA Tour players. Let's say your son's probability per par 3 hole is 1 in 3,000. A typical gold course will have about four par 3 holes. Let’s say your son plays every day. That would be 28 par 3 holes a week. The probability of making exactly two hole in ones would be combin(28,2)×(1/3000)2×(2999/3000)26 = 1 in 24,017.
Last week I was in Las Vegas playing Casino War. I was the only one at the table, with my girlfriend standing behind me watching. I was thinking about how much to bet, but I hadn’t placed a bet down yet, and the dealer started to deal my card. He then realized I had not put a bet down, and jerked it back, but did not burn it. I saw that it was a Jack, but I don’t think he realized I saw it. I was confused for a few seconds, waiting for him to burn it, and I was afraid of getting in trouble if I made a big bet, so I placed the minimum $10 and won the hand with the Jack over his lower card. In a situation such as this, a Jack isn’t a sure win, but would it have been legal/morally correct/with-in the rules of the casino to place a big bet down and keep the winnings? I wished I had placed a big bet down because the chances of winning with a jack are pretty high, but was afraid of getting some heat from a manager or security if I won a huge hand (no one was watching us though). What would you have done, or recommend to do in such a case?
Not that you asked, but you have a 43.4% advantage if your first card is a jack. It is the dealer’s fault for flashing the card. Contrary to what some members of the casino staff, especially in security, incorrectly believe, you are legally allowed to make use of whatever information made available to you under normal playing conditions.
Morally, you should follow your own conscience. You have to live your own life. That said, I think most players, including me, would be okay with increasing the bet in that situation. For one thing, game security is not the player’s job. For another, the casinos take advantage of, if not rely on, player mistakes. For example, consider the big 6/8 bet in craps. The casinos have no compunction about accepting a bet on that, when the place bet on 6 or 8 pays on exactly the same thing, but has better odds. See if you are offered forgiveness if you foul your hand in pai gow poker, even if the correct setting is totally obvious.
If it happens again, don’t get too greedy, and act nonchalant. If you suddenly go from a $10 to a $500 bet, it will set off all kinds of red flags. A good dealer would realize why, and ultimately the bet would not be accepted, or a card would be burned.
I am playing 8-5 triple bonus plus with a promotion adding $250 to each taxable jackpot. The double up feature is on the machines, and I am doubling each full house or better until I lose, or get over $1200. Can you assist in figuring the expected value on this game? Thanks.
Nice find! You didn’t say what denomination you are playing, which is important, so I’m going to assume dollars. For five-coin maximum bet, the number of doubles required for a win of w (where w<1200) is 1+int(log(1200)-log(w))/log(2).
The following table shows for each initial hand the pre-double win, pre-double probability, number of doubles required, post-double win, and probability achieving the post-double win, including the $250 bonus. The lower right cell shows a return of 115.5%. You will get a jackpot every 297 hands on average, with an average jackpot of $1,717.46.
8-5 Triple Bonus Return Table with $250 Bonus for Wins of $1,200 or More
| 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 |