Ask The Wizard #168
An increasingly common side-bet I’m seeing in Texas Hold ’Em games is for players to bet on the total "value" of the flop, where the value of a card is assigned via Blackjack rules. (ie, A=11, KQJ10=10, others are face) Players have the option to pick one or more total value numbers by putting a side bet in the pot. Play continues until the flop hits a value that is covered by a player (so, a flop of KK4 would send the pot to a player that bet on 24). Mathematically, what is the best number to bet on? Most games I have played in have the stipulation that 30 cannot be bet on, which would make you think it would be the "best" number to bet on, but considering 30 has a very limited number of ways it can be made (three 10-value cards, A-9-10, or A-A-8) I’m not sure that that’s true. Also, hands where more players have ten-value and Ace cards are more likely to see a flop. Would you be interested in doing an analysis on this side-bet?
I’m going to assume that if nobody wins on the flop that either the bets are refunded or the next flop is used to resolve them, as opposed to the turn and river cards being used. I am going to ignore the rule that if everybody folds then the bet is not resolved. Clearly this rule helps the lower totals but to factor that in the analysis would get complicated and subjective. That said, the table below shows the probability of each total. As you can see, the best bet would be on a total of 23, with a probability of 8.3982%.
Blackjack Points in Flop
| Total | Combinations | Probability |
| 33 | 4 | 0.000181 |
| 32 | 96 | 0.004344 |
| 31 | 504 | 0.022805 |
| 30 | 840 | 0.038009 |
| 29 | 784 | 0.035475 |
| 28 | 920 | 0.041629 |
| 27 | 1108 | 0.050136 |
| 26 | 1264 | 0.057195 |
| 25 | 1472 | 0.066606 |
| 24 | 1652 | 0.074751 |
| 23 | 1856 | 0.083982 |
| 22 | 1800 | 0.081448 |
| 21 | 1508 | 0.068235 |
| 20 | 1408 | 0.06371 |
| 19 | 1336 | 0.060452 |
| 18 | 1196 | 0.054118 |
| 17 | 1080 | 0.048869 |
| 16 | 896 | 0.040543 |
| 15 | 740 | 0.033484 |
| 14 | 512 | 0.023167 |
| 13 | 352 | 0.015928 |
| 12 | 268 | 0.012127 |
| 11 | 200 | 0.00905 |
| 10 | 136 | 0.006154 |
| 9 | 92 | 0.004163 |
| 8 | 48 | 0.002172 |
| 7 | 24 | 0.001086 |
| 6 | 4 | 0.000181 |
I work in a casino and have a bet that says a roulette dealer cannot influence the outcome of a roll. There are definitely those who think it can be done. Not to a number of course, but perhaps a section of the wheel. What would you consider a good test to reasonably determine whether a dealer has influenced the results? Assuming the number of trials is reasonable for us to attempt I will gladly share the results.
I’m on your side. If this could be done then dealers could easily conspire with players and share in the profits. Yet I never hear of this happening. A good test would be to get somebody who claims to be able to influence the roll and have him attempt to land it in a particular half of the wheel as many times as possible over 100 spins. The more times he makes it the greater weight his claim will have. The table below shows the probability of 50 to 70 successful spins. For example, the probability of 60 or more successful spins is 2.8444%. Common confidence thresholds in statistics are the 90%, 95%, and 99% levels. To beat a 90% confidence test, in which the probability of failing given random spins is 90%, the number of successful spins would need to be 57 or more. To beat a 95% test the number would need to be 59 or more, and at 99% the number would need to be 63 or more.
Probability of at Least 50 to 70Successful Roulette Spins
| Wins | Probability |
| 70 | 0.000039 |
| 69 | 0.000092 |
| 68 | 0.000204 |
| 67 | 0.000437 |
| 66 | 0.000895 |
| 65 | 0.001759 |
| 64 | 0.003319 |
| 63 | 0.006016 |
| 62 | 0.010489 |
| 61 | 0.0176 |
| 60 | 0.028444 |
| 59 | 0.044313 |
| 58 | 0.066605 |
| 57 | 0.096674 |
| 56 | 0.135627 |
| 55 | 0.184101 |
| 54 | 0.242059 |
| 53 | 0.30865 |
| 52 | 0.382177 |
| 51 | 0.460205 |
| 50 | 0.539795 |
In Reno there is a new type of positive EV promotion. The dealer pushes all “dealer draw to 21s”. Dealer naturals still win. (Any strategy suggestions? Table limits are $5-$25, I always play the max. The basic game is 6-Deck H17 DAS RSA to 4 hands.
Wow! According to my calculations this results in a player advantage of 6.4%. I’m assuming that the rule applies after doubling and splitting. Here is the basic strategy for that rule.
I’ve been unable to find a risk-of-ruin table for full-pay pick ’em poker or related info about its variance. The $1 full-pay machines clearly have the best returns in the buffalo-niagara region, but i’m uncertain about suggested bankrolls. Any advice would be much appreciated. Thanks.
The standard deviation in Pick ‘em Poker is 3.87. The standard deviation in conventional video poker tends to run from about 4.4 to 6.4. I don’t have any risk of ruin tables for Pick ‘em Poker. So the best advice I can offer is to use the jacks or better table in my video poker appendix 1. Jacks or Better has the lowest standard deviation in that appendix at 4.42, so you can be a little more aggressive than that table calls for.
In my regular home game, players often call many different wild games. Usually, there will be 2 wilds (baseball, follow the queen where both the queen and the next card are wild, football) and occasionally only 1 (our version of 3-5-7, follow the queen with only the next card wild.) In these games with 4-8 wild cards potentially out there, which is statistically less likely? 5 of a kind or a straight flush? There is constant argument over this and I would love for a reputable and universally respected source such as yourself to settle the matter. Thank you in advance.
The five of a kind is less likely. I just added a table to my section on poker probabilities detailing the probability of each hand according to each individual rank as wild.
Hi wiz. First I love the site. You are turning me into a Spanish 21 Player. MY wife and I hit Laughlin once on a trip to LV. I enjoyed the vibe. Why can’t I find any good sites on Laughlin? We play BJ, S21 and Craps. We are going to NV in September and would enjoy stretching our playing budget. Where do you play when your on the river?
When in Laughlin I prefer Harrah’s. Although it is also the most expensive in my opinion, it is worth the extra money. I find the service at every other casino to be slow and poor, and the median age of the clientele to be about 65. However, when I get in the mood for something less corporate and polished I head to the Riverside, the only family owned casino in town.