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Ask The Wizard #61
In an 8-deck baccarat game, what are the chances of both the player and banker being dealt an Ace and an 8 of Diamonds in the same hand?
(82/combin(416,2))* (72/combin(414,2)) = 0.00000043, or 1 in 2308093
Is there a betting strategy for baccarat that involves progressive betting? Are there particular websites dedicated to this?
There are numerous options available, and unfortunately, they're mostly ineffective.
I absolutely adore your website! It's fantastic. I have a question about your discussion on 'robot players' in online casinos. You mentioned that: 'These robots require significant skill and time to operate, but if utilized correctly, they can transform a computer into a profitable enterprise,' and that this is why casinos sometimes withhold payouts. What puzzles me is your insistence, similar to any statistician, that ultimately everyone loses over time. Therefore, how can a robot change the outcome? Why would this be a concern for casinos? Even if they play perfectly, the house still maintains the upper hand, right? In relation to expected results in blackjack, I found your tables in Appendix 4 about standard deviation to be quite beneficial. However, I’m curious about the likelihood of experiencing a downturn right from the start of gameplay, and not recovering to an average of 100% (after considering losses from the house edge)? What might the probabilities look like over various hands such as 100, 200, 1000, 10,000, and 100,000?
Lastly, could you clarify why it's a misconception that winning becomes more likely after a series of losses? From my perspective, since the expected return is around 99.5%, if after 1000 hands your return is at 78%, wouldn't that imply that a winning hand is more likely on the next play? People often say that cards 'lack memory,' but isn't it true that the natural distribution effectively serves as a form of memory? I would greatly appreciate your insight on this topic! Thank you very much.
You're correct that utilizing a robotic player in a standard game might lead to greater losses. However, some casinos do present games that offer a player advantage when played skillfully. For instance, Unified Gaming had a blackjack game that allowed a player edge of 0.5% for several months, although that is no longer the case. Many casinos using Real Time Gaming software provide a joker poker game with an expected return of 100.18%. Additionally, several casinos have promotions where the player who participates in the most hands within a set timeframe can win a reward, where a robot player would clearly benefit. Regarding your second question, the bell curve represents a predictive model for the combined results of numerous random events. It's important to recognize that past occurrences cannot influence future results. Once an event happens, it becomes a definite outcome rather than a variable. If you played 1000 hands of blackjack resulting in a return of 78%, it means you experienced an unfavorable result within the tail end of the bell curve during that period. From the 1001st hand onward, your results could fall anywhere on a new bell curve. I hope this clarifies your doubts, but to gain a more thorough understanding, a solid course in statistics would be necessary.
Wizard - What is the benefit of playing two hands in Three Card Poker instead of one, if there is any? Some casinos allow for two hands to be played, starting with one before making a decision on the second. Others permit only the second hand to be played without looking, which likely does not favor the player. Thank you!
Great question! In Stanley Ko’s handbook Mastering Three Card Poker, he indicates that having knowledge of the first hand lowers the house edge from 3.37% to 3.31% on the second hand when utilized correctly with a concealed computer. Even if you could observe all seven hands at the table, the house edge would still be 2.32%.
In blackjack, what are the chances of achieving a blackjack?
The probability varies with the number of decks used. If there are n decks, the formula for probability becomes 2*pr(ace)*pr(10) = 2*(1/13)*(16*n/(52*n-1)), which conveniently rounds off to about 1 in 21.
Here’s the precise probability calculated for different deck counts.
Probability of Blackjack
| Decks | Probability |
|---|---|
| 1 | 4.827% |
| 2 | 4.780% |
| 3 | 4.764% |
| 4 | 4.757% |
| 5 | 4.752% |
| 6 | 4.749% |
| 7 | 4.747% |
| 8 | 4.745% |
What is the likelihood of playing ten hands and not achieving a (two-card) 21? Assume the cards are reshuffled after each round.
If the probability of getting a blackjack is p, then the probability of not hitting any blackjacks in 10 hands follows the formula 1-(1-p). For example, in a six-deck game, the answer would be 1 - 0.952511.10What are the chances of getting three blackjacks consecutively in a situation with one deck, four players, and one dealer?10= 0.385251.
Assuming there’s no shuffle between hands, the outcome is unaffected by the other players. The calculation gives us 2 * (16/52) * (4/51) * (15/50) * (3/49) * (14/48) * (2/47) = 0.00004401, which approximates to 1 in 22,722. If there is a shuffle after each hand, the probability significantly increases.
First of all, I want to express my appreciation; your website is truly exceptional. Thank you! I recently observed a new craps game called 'Four The Money' being played at Grand Casino in Biloxi, MS. To win, the shooter must roll the dice four times without rolling a 7. What are the probabilities of rolling the dice:3Could you explain how the mathematics behind this works? Thanks!
You're welcome! I appreciate your kind remarks. The probability of rolling the dice n times without rolling a 7, followed by rolling a 7, is expressed as (5/6)^(n) * (1/6). For just rolling n non-sevens, without specifying what the next roll would be, it would be (5/6)^(n). Hence, the probability of rolling the dice at least four times without hitting a seven is calculated as (5/6)^(n).
4 times without throwing a 7?
3 times without throwing a 7?
2 times without throwing a 7?
1 times without throwing a 7?
I am curious to know if the Las Vegas Video Blackjack reshuffles the cards after each hand or only after playing through all the decks. I understand that in Las Vegas, the tables shuffle after all decks are played because shuffling after every hand would discourage participation. Does shuffling after each hand worsen your winning odds? Is this practice even permissible or necessary?
I'm not certain about the timing of shuffles, but I would suggest it happens after each hand. From my understanding, you will find that player odds slightly improve if the dealer plays a precise number of hands between shuffles (including one), rather than concluding a hand and then shuffling immediately.nHere, you can find mathematically sound strategies and insights concerning casino games such as blackjack, craps, roulette, along with hundreds of other games you can engage in.nPlease check your email and click the link we sent to finalize your registration.4=625/1296=0.4823.
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