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Every time I go to the Blackjack games there is a grumpy simple individual, who wants to stone some poor soul for "messing up the shoe.” Is there any truth to this?

Jim from Las Vegas

In ten years of running this site I steadfastly denied the myth that bad players cause other players to lose in blackjack. However, you are the lucky 1000th person to ask, so I took the trouble to prove it by random simulation. The rules I put in are the standard liberal Vegas Strip rules as follows.

6 decks
Dealer stands on soft 17
Double on any first two cards allowed
Double after split allowed
Late surrender allowed
Player may re-split to four hands, including aces
Cut card used

First, I had both players follow correct total-dependent basic strategy. Over almost 1.6 billion rounds, the loss of the first player to act was 0.289%, and the second player to act of 0.288%.

Second, I had the first player follow the same correct strategy, and the second player follow the same correct strategy except:

Always hit 12 to 16
Always double 9 to 11
Split any pair
Never surrender
Never soft double

In a simulation of 1.05 billion hands the loss of the first player was 0.282%, and the second player was 11.260%. So the house edge of the basic strategy playing first player was almost the same, regardless of whether the second player played correctly or wildly incorrectly. I hope this puts and end the third baseman myth, but I doubt it. As I have said many times, the more ridiculous a belief is, the more tenaciously it tends to be held.

I was reading the Blackjack entry on Wikipedia, and it said that when blackjack was not that popular, casinos added a bonus payout of 10 to 1 for the ace of spades and any black jack, hence the name. It goes on to say that the bonus was quickly removed, but the name stuck. I was wondering what the house edge was with a 10-1 payout for that hand.

In a six-deck shoe, the probability of such a blackjack is 2 × (6/312) × (12/311) = 0.001484. I’m going to assume that if the dealer has a blackjack also, then the hand is a push. That said, the probability of the dealer not getting blackjack, given that the player did, is 1 - 2 × (23/310) × (95/309) = 0.954379. So the probability of winning such a blackjack is 0.001484 × 0.954379 = 0.001416. The value of an additional 8.5 units whenever that happened is worth 8.5* 0.001416 = 1.2039%. Assuming otherwise liberal Vegas Strip rules, with a house edge of 0.28%, the player edge with the 10 to 1 rule would be 0.92%.

If one plays a Jacks or Better video poker machine, at 40,00 hands per session with perfect strategy, I assume that a royal would appear about every 10 sessions. What are the odds of not hitting a royal for an entire year (about 50 sessions playing once a week)? Thank You.

Dave S. from New Haven

I assume that you assume the probability of a royal is 1 in 40,000. Playing 4,000 hands per session the expected number of royals per session is 0.1. A very close appoximation for the probbility of zero royals per session is e-0.1 = 90.48%. The reason it is not 90% is because sometimes you will get more than one royal per session. The expected number of royals in 50 sessions is 0.1 × 50 = 5. The probability of zero royals in 50 sessions can be closely approximated at e-5 = 0.67%. The exact probability is (39,999/40,000)^(200,000) = 0.67%, as well.

I’m learning pai gow (tiles) for my next trip to Vegas. I would like to take advantage of the "banking" rules, but I have a couple of concerns, specifically banking and comps. If another player at the table shoves in a huge bet after I decide to bank, can I change my mind? Also, what happens if I don’t have enough chips to cover all the bets at the table?

Second, I hear that pai gow players are rated much lower than other table games per bet due to the slower pace of the game, true? Do you have an educated guess what the average bet in pai gow would have to be to equal the comp privileges of a \$25/bet Blackjack player?

Uncle Mo from Parker

Your timing is good with this question. The summer of 2007 shall be referred to as my "summer of pai gow," because I am devoting a great deal of analysis to the game. About banking, yes, you can change your mind about banking, if a player makes a bet larger than you are comfortable with. It is a rule that you must have enough chips in front of you to cover all the action. My May 5, 2007 column shows that pai gow is rated at 30 hands per hour, at least by the casino that gave me the figures. If you multiply the assumed house edge and hands per hour in that table, you can see the blackjack is rated at 0.525 bet units per hour, and pai gow at 0.495, so blackjack is only slightly better at the same bet size. Although the expected loss is greater in pai gow, the standard deviation is much less. This makes pai gow a great game if you are playing for a rating, and wish to minimize risk.

First of all, thanks for the wonderful site. I recently saw a set of bonus bets in Baccarat called 4-5-6, on the total number of cards between the player and banker hands. The odds they offer at the Atlantic City Hilton are 3 to 2 for 4 cards, 2 to 1 for 5 cards, and I believe 3 to 1 for 6 cards if I remember correctly. That means we should see more hands that end with 4 cards. What are the odds on all three bets?

Ray from Egg Harbor Township

You’re welcome, thanks for the compliment. Without knowing anything about the probabilities, if those were the payoffs, then there would exist a player advantage on at least one bet. The way you can tell is to take the sum of 1/(1+x), where x is what the bet pays on a "to one" basis, over all the bets. If this sum is less than 1, then at least one bet has a player edge. In this case, according to your odds, this sum would be 1/2.5 + 1/3 + 1/4 = 0.9833. This trick may come in handy, for example, if you see an amateur putting up sports betting futures.

What is probably the case here is that six cards pays 2 to 1. Based on that assumption, and six decks, the house edge is 5.27% on four cards, 8.94% on five cards, and 4.74% on six cards. For more information see my baccarat appendix 5.