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Do you feel it's better to play one slot machine or many, and how much should I feed before walking away. If given a touch screen to stop tumblers should I stop them.

Gene from Laguna, USA

None of these factors matter. Walk away when you're not having fun any longer.

I understand the calculations made to determine the house advantage for the various games. My question is most applicable to the games with a large payout (for example, a royal flush in Let It Ride). Would it not be more reasonable to calculate the "house advantage" with the royal flush excluded? While hitting the royal is possible, it is extremely unlikely for the average player. Would this modified house advantage be more applicable to the average player? Thanks for the great site.

Bill from Corpus Christi, USA

You make a good point. In terms of what to expect in the short run then you should ignore the highest hands. I know video poker players sometimes disregard royal flushes when determining their short-term expectations. However, as a mathematical purist, I can't help but consider every possible outcome, regardless of how unlikely.

I love to play craps and would like your opinion on a conventional method of play. Pass line and two come bets with full double odds or with one come bet? Does having three different bets working superior to two?

Richard from Binghampton, USA

As long as you are backing up your pass and come bets with full odds, it doesn't make any difference how many come bets you make. However, it does reduce the overall house edge to keep the odds on your come bets working on the come out roll.

Here in Netherlands we have also Caribbean Stud Poker. The progressive jackpot side bet payoff table is same as "Table 3" but the straight flush pays always \$5,000 instead of 10% of the Jackpot. How do I calculate the break even point?

Jan from Rotterdam, Netherlands

According to table 3, a four of a kind pays \$500, a full house pays \$100, and a flush pays \$50. If m is the amount of the jackpot meter then the return per dollar bet is (1121800+4*j)/2598960. The meter would need to reach \$369,290 for this to be a positive expectation bet.

What is your opinion of online progressive slots, such as the Dazzler at Unified Gaming casinos? The jackpot starts at \$15,000. If the jackpot gets large (> \$75,000), do you think the player has an edge?

Mark from Allston, Massachusetts

I don't have any information on the odds of that game so I can't tell you at what point the meter is high enough to give the player an edge. However, keeping an eye on the meter and only playing when it is high is a good idea.

Hey, Wiz. I am curious to know what the expected number of spins is for an American Roulette wheel before all 38 numbers have been chosen at least once. Is this proportional to the number of selections (38) or is it exponentially related to this number? I tried figuring this out for a 6-sided die but got stuck fast.

Scott from Elmhurst, Illinois

Once you have hit n numbers, the probability of getting a new number on the next spin is (38-n)/38. If the probability of an event is p, then the expected number of trials before it happens is 1/p. Thus the expected number of spins to get a new number, given that you already have n, is 38/(38-n). For example once you have hit 20 numbers the expected number of spins to get the 21st is 38/18=2.11. So the answer is the sum of the expected number of spins at each step: (38/38)+(38/37)+(38/36)+...+(38/1)=160.66.

I have been mulling over some money management techniques, and wanted to thank you for some sound advice (more to the point, probability theory) and it seems that win probabilities are based roughly on your starting bankroll. For example, estimating the probability of winning \$100 on a \$200 buy-in without going broke first. This is very helpful, but without doing the math (shame on me, I know), I get the feeling win limits should be based more on your betting unit, i.e. \$1, \$5, \$10, etc. Basically, the idea you will have smaller fluctuations over time using a smaller bet than a larger one. I guess my actual question is this: if I have a given bankroll, (say \$100) and a stated win limit (say \$50), what (if any) betting unit would give the greatest possibility of success? I am thinking too small of a bet would give lower my chances of getting much above the mean and too large would run the risk of bankruptcy. Any advice or comments?

Scott from Saline, Michigan

In your example, assuming a negative expectation game, the best bet size to maximize the probability of reaching your win goal is \$50. In a positive expectation game the best bet size is as small as possible. The reason is that the more you play the more the house edge will grind you down, or the more you will grind the casino down if you have the edge.

Would it be possible to use a Java decompiler to look at the source code of the Unified Gaming blackjack applet (and determine whether the game is fair or not)?

Mark from Allston, Massachusetts

I would hope they put enough encryption on their game to make that difficult. Nevertheless, I wouldn't say it is impossible.

As a BJ player I regularly split 10s vs. bust cards. This move has a positive expectation, but isn't as profitable as standing pat. Could you please show this friend of mine how much worse splitting 10s is than standing pat, in a neutral deck? Also, lots of BJ players get upset at someone who splits 10s vs. 6, but they often split 2s or 7s vs. 10 themselves. Can you set the record straight on this?

Brett from Richland, USA

I took great pains to create my blackjack appendices 9A-9H to answer these kinds of questions. For example, in a six-deck game where the dealer stands on soft 17, you would use blackjack appendix 9G. There you can see the expected return by standing on two 10's against a dealer 6 is 0.702826. The expected return by splitting in this situation is 0.622165. So, the player can expect to win an extra 8.07% by standing as opposed to splitting. Don't pay much stock in what other players say or do at the blackjack table.

What would be the best way for one to win a blackjack tournament. I seem to do quite well in regular play, but can never come out in the top two to advance. It seems third place is the best I can do.

Helene from Sherman Oaks, USA

One could write an entire book on blackjack tournament strategy. Briefly, here are some pieces of advice:

1. Positional advantage is very important. When you are last to act is the best time to take chances with big bets.
2. Bide your time at the beginning. Sometimes on a cold table everyone else will burn themselves out while you coast to first place at your table.
3. The second half of the round take big chances to get in first place.
4. If competing against one other player you want to bet with him when ahead, and contrary to him when behind.
5. Pay attention to the maximum bet allowed. If the maximum bet is small compared to the player stacks you should get aggressive early.