Ask The Wizard #123

Indeed, that happens quite often. It's common for high-stakes players to receive a 10% refund on their losses. Personally, I believe this is a high-risk practice, as savvy players could easily take advantage of this system to tip the odds in their favor. The optimal candidate for this offer would be someone who plays extensively in games with a high house advantage. Conversely, those who engage in games with lower house edges for brief periods and varying bet sizes are the ones who could really benefit. It may sound contradictory, but under this arrangement, players need to incur losses to reap any rewards. Consequently, players should aim for a significant winning target while keeping their loss limit relatively modest. If we set a hypothetical winning target at $1,000,000 and a loss limit at $100,000, the chances of achieving the goal would be approximately 1 in 11, which I'll elaborate on later. After factoring in the 10% rebate, the expected value becomes (1/11)*$1,000,000 + (10/11)*(0.9*-100,000) = +$9091. An effective approach for quickly reaching a lofty win goal could involve a strategy similar to an anti-martingale, where players increase their bets following wins.

In my attempts to disprove betting systems, I used to argue that previous outcomes are irrelevant in gambling. However, I occasionally faced pushback from those who pointed out that past results are indeed significant for card counters, which is a valid point. Thus, I now clarify that in independent trial games such as roulette and craps, historical data does not influence outcomes. baccarat appendix 2 A shoe containing more low-value cards is advantageous for players, while a deck rich in high-value cards benefits the banker. Therefore, in baccarat, there exists an extremely minimal tendency for the next outcome to deviate from the previous result. Thus, while the odds do fluctuate during the game as cards are drawn, these changes are trivial. For practical purposes, the game does not lend itself to card counting. I can't definitively say whether the banker could win every hand, but I would speculate that it's possible.

Neglecting the house edge, the likelihood of hitting the winning goal is calculated by taking the loss threshold divided by the total of the loss threshold plus the winning threshold. Here, that's 1000/(1000+100) = 1000/1100 = 90.91%. However, this probability will be negatively affected by the house edge of the specific game being played, as well as the size of each bet; smaller bets generally correspond to a diminished chance of meeting the winning goal.

My charge for conducting a straightforward test would be $2000, reflecting the value of my time for this task. Offering a reward of $20,000 if you succeed in the challenge costs almost nothing for me, as the probability of winning is virtually negligible.

I would treat them all the same. Typically, standard three-reel slots are configured to provide a similar return, depending on the specific casino and the denomination of the coins used.

To determine this, if the chance of an event is p, then the expected number of trials until it occurs is 1/p. Let’s denote x as the expected number of rolls for a shooter. The likelihood that a round will finish in just one roll (with outcomes of 2, 3, 7, 11, or 12) is 1/3. If a 4 or 10 is rolled on the come-out, the anticipated number of subsequent rolls is 4, given that the chances of rolling a 4 or 7 stand at (6+3)/36 = 1/4. Similarly, if a player rolls a 5 or 9, the expected number of additional rolls drops to 3.6, while for a 6 or 8, it averages out to 36/11. Assuming a point has been made, the probabilities for 4 or 10, 5 or 9, and 6 or 8 are 3/12, 4/12, and 5/12, respectively. Hence, the expected number of rolls per round becomes 1 + (2/3)*((3/12)*4 + (4/12)*3.6 + (5/12)*(36/11)) = 3.375758. Furthermore, the odds that the player will seven out is (2/3)*((3/12)*(2/3) + (4/12)*(3/5) + (5/12)*(6/11)) = 0.39596, and thus the chance of not sevening out is 1 - 0.39596 = 0.60404. So...

x = 3.375758 + 0.60404*x
0.39596*x = 3.375758
x = 8.52551

That’s a thought-provoking question. If the counter is generous with tips, the dealer has the choice of either remaining silent to continue receiving more tips or reporting the player to curry favor with casino management. Ultimately, it often comes down to the dealer's personal outlook — whether they support the players or are aligned with the casino's interests. Dealers who prioritize their employer are likely to report suspected counters, and tips may have minimal influence on their decision. Since tips are typically pooled, the dealer receiving your tip might see only a fraction of it. If a dealer is cynical and does not like how tips are shared, tipping may not yield much in the way of protection. My belief is that female dealers tend to be more loyal to the casino compared to their male counterparts, and among different ethnicities, Asian dealers reflect similar loyalty. One of the books I read about blackjack goes into a deeper discussion on this, though I can’t recall which. The issue of tipping remains controversial in card counting circles, and various counters follow the philosophy of Stanford Wong, opting to tip only when the cover gained exceeds the value of the tip itself. This might explain the humor that suggests the difference between a counter and a canoe is that a canoe sometimes tips over, while counters keep tipping regardless, believing in the practice.