What are the odds of getting a Royal Flush on a video poker game by holding two cards and drawing three to make the royal flush?

Jim K. from Laguna Niguel, California

The number of ways to draw 3 cards out of the 47 remaining in the deck is combin(47,3)=16,215. One of those will be the three needed for a royal so the odds are 1 in 16,215.

What is the house advantage on put bets with 20 times odds. Should the house allow put bets at these odds, for example someone could take \$1000 with \$20,000 on every number. Could you explain this to me? Thanks, Great site.

Bry from Chesterton, Indiana

For other readers, let me explain that a put bet is making a pass or come bet after a point has already been established. The player may choose the point to be established on the put bet. While the player can make an odds bet immediately on top of the put bet, the opportunity to win on the initial roll is lost. The effect is the same as making a place bet or buying odds, but the house edge is different depending on the multiple of odds allowed. In the case of 20 times odds the house edge of the put bet on the 4 and 10 is 1.59%, on the 5 and 9 is 0.95%, and on the 6 and 8 is 0.43%. At this high level of odds allowed, which is much greater than the norm, all put bets are better than the corresponding place or buy bets. This option should never be taken at a casino that offers less than 5 times odds. At 5 times odds exactly, the put bet on the 6 and 8 is slightly better than the place bet. At 10 times odds or greater, all put bets become better than their corresponding place or buy bets. I shall add something to my craps section about the put bets, thanks for the idea.

Across the various blackjack strategies you provide on your site, does the standard deviation change much? I would guess that when DAS is allowed, one's results are more variable, but how much? Is there a good reference book that provides variance figures for various basic blackjack strategies?

Nathan from University, Mississippi

Good question. Unfortunately I don't have exact standard deviation figures according to specific sets of rules. The 1.15 figure on my site is based on liberal Vegas Strip rules. I agree that the double after split rule increases the standard deviation. Surrender would decrease it. Sorry I am not of more help than that.

I know this will be almost impossible for you to figure out, but I'm curious to know approximately how many people play Cash Splash on a daily basis and some idea of the odds against me. I am especially curious to know if online progressive slots offer better chances of hitting that jackpot than land-based progressives.

Donna from Los Angeles, California

You're right, it is impossible for me to know without Microgaming giving me the details on how their reels are weighted. I have asked some of the major software companies for such information, but thus far nobody has volunteered anything. However, I can tell you that the average payback for all slots at the Golden Palace for the month of March 2000 was 95.67%. This information is available at the Golden Palace web site, click on the Price Waterhouse Coopers monthly payout review.

Hello Mr. Wizard. I've been visiting your site for almost a year and want to take this opportunity to thank you. Thank you not only creating the site, but for constantly updating it with new and informative material. I can only imagine the amount of time your must put into it.

I have a number of questions. The first is about the banners. I'd like to know how we could optimize your income by clicking on the banners. Do you get paid on the basis of hits, unique hits or a flat fee or some other formula? Currently I only click on a banner if it's something new, but I'd be happy to click on a few banners every time I visit, if that would help generate income.

The next questions I have are about your perfect strategy for Jacks or Better table, and the practice Jacks or Better game. I hate to seem dense, but that's never stopped me from asking questions before. Where on the table do you find the rank of a hand with 1 high card (A,K,Q,J) and no penalty cards? My second question is somewhat related to the first. According to the practice game, the optimum play for a hand with unsuited A,Q,K and no penalty cards is to hold the K,Q and discard the A with the low cards. Intuitively, I would have thought that keeping the Ace is the better play. What is the advantage of dropping the Ace, and how would I determine the optimum play on this hand from the table?

Denis from Rochester, New York

Thanks Denis for your kind words and your interest to help keep the site financially healthy. When you first wrote just clicking my banners helped. However, as I update my answer in 2013, it is acquisitions that put rice on my table. It doesn't do much good to click on the banner if you don't at least sign up for an account, and preferably make a deposit.

You can see from my 9-6 jacks or better strategy, that there is a single line for "one high card." That is because an individual jack to ace is worth about the same.

On your blackjack page, in the House Edge section, you indicate that the player has the edge in single-deck blackjack. Numerous Vegas Casinos offer single-deck blackjack, but I doubt they are dealing a losing game. What gives here? Is your page incorrect or are the casinos just gambling on their clients not being able to play basic blackjack. A friend of mine and I have a bet riding on this. Also. Is it true that with even a simple card counting system a competent blackjack player can consistently walk away a winner in the long run? Thanks for your help.

You have a legitimate point. At the time you wrote this I indicated the house edge according to a fixed set of rules that varied only by the number of decks. However, in real life, when single deck is offered the other rules invariably become more stingy. I have already enhanced the house edge table in the blackjack section to include a wider variation in rules. The typical Las Vegas single-deck game does not allow doubling after a split and the dealer hits a soft 17. With these rules, the house edge is 0.18%. The best single-deck game is at the Slots-a-Fun (next to Circus Circus) where the dealer stands on soft 17 for a player edge of 0.01%. Next door, at the Westward Ho, double after a split is allowed but the dealer hits a soft 17 for a house edge of 0.04%. About card counting, absolutely not, even the best of card counters will lose often. An entire month can be at a loss. Only over the very long haul does a net profit become likely.

Update: Since this writing the Westward Ho was torn down and the Slots-a-Fun has slots only.

Dear Wizard! I'm about to play Blackjack on Finnish site with the following rules:

• 4 decks
• 9-11 double
• one split only
• double after split allowed
• one card to split aces
• dealer stands on soft 17
• European card rule= none
• no surrender

Are these rules a good deal for me? What is the house edge? Thank you for an answer.

Kim from Helsinki, Finland

The house edge in this game is 0.51%.

What are the odds of being dealt a Royal Flush? Of being dealt a SEQUENTIAL Royal Flush (forwards or backwards)?

Ric from Torrance, California

The probability of any royal flush is the number of possible royals, which is four (one for each suit), divided by the number of ways to choose 5 cards out of 52, which is combin(52,5)=2,598,960. So, the answer is 4/2,598,960 = 0.00000153908, or 1 in 649,740.

The probability of a sequential royal flush equals (number of suits) * (number of directions) / (total permutations of 5 cards out of 52) = 4 * 2 / permut(52,5) = 8 / 311,875,200 = 8 / number of possible royals, which is four (one for each suit), times the number of directions it can be in divided by the number of ways to pick 5 cards out of 52, which is permut(52,5)=311,875,200. So, the answer is 4/311,875,200 = 0.00000002565, or 1 in 38,984,400 .

I was wondering how to alter dice in the game of craps so it hits 7 or 11 every time can you help. thank you.

d from Monterey, California

Alter then so that one die has a six on every side, and the other one has all ones and fives.

I'm a college drop-out. I was studying for a mathematics degree, but earned an MCSE and made more money. But I LOVE statistics, and currently work for a consulting company in an Indian casino. I've never heard of actuaries until I saw this site. How does one become an actuary?

Dave from Redding, California

To become an actuary one must pass several examinations. The first set of exams are mathematically intensive and the second set are based on the theory of investments, actuarial science, pensions, and other such topics. However it will be difficult to find employment without a college degree. Most actuaries have at least an MA in mathematics. Your Microsoft certification would certainly give you bonus points but not enough to compensate for a lack of a degree. To learn more visit the Society of Actuaries web site.

Update: Since this question was posted the Society of Actuaries changed the exam requirements. There are now fewer but harder exams. That is about all I know about it.