# It's Mathemagic!

A couple of weeks ago, a colleague discussed with me the possibility of creating a simple three card sidebet for a game he was working on.  Essentially, it was no different than the Pair Plus wager for Three Card Poker. As a favor, I quickly gave him the frequencies for the hands and made sure there was a note that said that these applied ONLY to a standard 52-card deck.  The next day I received a frantic voice mail that stated that he wanted these numbers for a blackjack-style shoe and what did I mean that the numbers only applied to a single deck.  Since there is still the same number of cards of each type relative to the others, shouldn’t the frequencies stay the same?

This led to a lengthy conversation in which I tried to explain why the frequencies change as the number of decks goes up.  Generally speaking, the frequency of sets (i.e. Pairs, Trips, etc…) goes up while the frequency of runs (Straights) goes down.  Below is a table that shows the frequencies of the winning hands in Pair Plus for both a single deck and a 6-deck shoe.

 Hand 1-deck 6-decks Mini Royal 0.0181% 0.0172% Straight Flush 0.1991% 0.1896% Three of a Kind 0.2353% 0.5248% Straight 3.2579% 3.1021% Flush 4.9593% 5.8424% Pair 16.9412% 19.4918%

The table clearly shows that which I just stated.  The frequency of a Three of a Kind more than doubles when we move to 6 decks.  A pair occurs about 15% more often.  Flushes occur about 20% more often.  In the meantime, Straights decrease by a mere 5%.  Our Royals and Straight Flushes decline slightly as well as they are a blend of Straights and Flushes.  The overall hit frequency goes up about 15% from 25.6% to 29.2%.  Is it any wonder that a paytable that works well for one deck would be a disaster (for the casino) with six decks?   If we were to take the standard Pair Plus paytable and apply it to a 6-deck version, the payback would go from just under 98% to nearly 115%!

My colleague was having a tough time believing me, and I did my very best to try and explain it.  I used the Three of a Kind as an example.  Once the Player is dealt a Pair, he is looking for 1 of the 2 remaining matching cards with 50 remaining in the deck for a single deck game.  Thus he has a 4% chance of drawing the Three of a Kind at that point (2 out of 50).  When dealing from a 6-deck shoe, there are still 22 cards of that same rank left in the shoe out of 310 cards.  This gives the Player a better than 7% chance of drawing a Three of a Kind from a Pair.  With a single deck, the probability of drawing additional cards of the same rank decreases greatly as each card of that rank is dealt out.  Whereas, with a large shoe, the probability of drawing cards of the same rank does not decrease nearly as much as each was in used up.

Taking this to the extreme, imagine a game where 4 cards are dealt and you are trying to draw a Four of a Kind.  In a single deck game there is are only 13 ways to do this.  By the time you have been dealt Three of a Kind, you have only a 1 in 49 chance of being dealt the fourth card of the same rank.  In a 6-deck version, you would a 21 in 309 chance – which gives you a 3 times greater chance of hitting it from that point.  In total, you would have 13/270,725 (0.0048%) probability of getting a Four of a Kind from a single deck vs. a 0.0357% probability from 6 decks.  This translates to nearly 7 ½ times as likely to get it using six decks.

There are many lessons that can be learned from all this.  The first is a simplistic lesson in why casinos like large shoes for games like blackjack.  It is much harder for a Player to count cards and trying to figure out what is coming up.  Just because a few small cards have been dealt doesn’t mean large cards are that much more likely than normal to show.  With a single deck, just a few small cards means the probability of large cards increases by a far larger margin.

The second lesson is that it is not always that easy to compare paytables from one game to another.  If a casino were to offer a Pair Plus type wager from a large shoe, they would have no choice but to offer lower payouts than they do for a single-deck game.  However, that would not necessarily mean that the overall payback is lower.  You need to make sure that you are always compare apples to apples.  A game dealing 7 cards can’t offer the same payouts as one dealing 6 for like poker hands, but you can’t just look at the payouts and say that one is ‘worse’ than the other.  You have to look at the actual frequencies of each winning hand and determine the true overall payback.