## Randomization

There are exactly 52 factorial (expressed in shorthand as 52!) possible orderings of the cards in a 52 card deck. In other words there are 52 × 51 × 50 × 49 × ··· × 4 × 3 × 2 × 1 possible combinations of card sequence. This is approximately 8×10^{67} possible orderings or specifically 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000. The magnitude of this number means that it is exceedingly improbable that two randomly selected, truly randomized decks will be the same. However, while the exact sequence of all cards in a randomized deck is unpredictable, it may be possible to make some probabilistic predictions about a deck that is not sufficiently randomized.

### Sufficient number of shuffles

The number of shuffles which are sufficient for a “good” level of randomness is a fundamental question, and the answer depends on the type of shuffle and the measure of “good enough randomness”, which in turn depends on the game in question. Broadly, for most games, four to seven good riffle shuffles (GRS) are both necessary and sufficient: for unsuited games such as blackjack, four GRSs are sufficient, while for suited games with strict conditions on randomness, seven GRSs are necessary. There are some games, however, for which even seven GRSs are insufficient.

In practice the number of shuffles required depends both on the quality of the shuffle and how significant non-randomness is, particularly how good the people playing are at noticing and using non-randomness. Two to four shuffles is good enough for casual play. But in club play, good bridge players take advantage of non-randomness after four shuffles, and top blackjack players supposedly track aces through the deck; this is known as “ace tracking”, or more generally, as “shuffle tracking”.

### Research

Following early research at Bell Labs, which was abandoned in 1955, the question of how many shuffles was required remained open until 1990, when it was convincingly solved as *seven shuffles,* as elaborated below. Some results preceded this, and refinements have continued since.

A leading figure in the mathematics of shuffling is mathematician and magician Persi Diaconis, who began studying the question around 1970, and has authored many papers in the 1980s, 1990s, and 2000s on the subject with numerous co-authors. Most famous is (Bayer & Diaconis 1992), co-authored with mathematician Dave Bayer, which analyzed the Gilbert-Shannon-Reeds model of random riffle shuffling and concluded that the deck did not start to become random until five good riffle shuffles, and was truly random after seven, in the precise sense of variation distance described in Markov chain mixing timae; of course, you would need more shuffles if your shuffling technique is poor.Recently, the work of Trefethen et al. has questioned some of Diaconis’ results, concluding that six shuffles are enough. The difference hinges on how each measured the randomness of the deck. Diaconis used a very sensitive test of randomness, and therefore needed to shuffle more. Even more sensitive measures exist, and the question of what measure is best for specific card games is still open. Diaconis released a response indicating that you only need four shuffles for un-suited games such as blackjack.

On the other hand variation distance may be too forgiving a measure and seven riffle shuffles may be many too few. For example, seven shuffles of a new deck leaves an 81% probability of winning New Age Solitaire where the probability is 50% with a uniform random deck. One sensitive test for randomness uses a standard deck without the jokers divided into suits with two suits in ascending order from ace to king, and the other two suits in reverse. (Many decks already come ordered this way when new.) After shuffling, the measure of randomness is the number of rising sequences that are left in each suit.

## Casino fined due to illegal blackjack play

# When a person is banned from playing at a casino, the casino..

must protect the player. Most casinos across the country have a list and if you are included for some reason by the casino or if you place yourself on the list, the casino must make sure that you do not gamble. The Revel Casino in New Jersey recently had to pay a large fine due to letting two men who were blocked from game play to play the game of blackjack.

The New Jersey Division of Gaming Enforcement filed a complaint which stated that the casino allowed two men to play blackjack from July to August of last year even though the two men were on the banned gamblers list. The casino must now pay **$27,500** due to the incident.

In total, the casino must now pay** $37,500** in fines for four different charges for this month. The violations include the blackjack charges and failing to follow the rules for the table game drop boxes collection. The casino has yet to comment on the incurring fines.

The largest fine in the bunch is from the two men who were able to take part in the gaming without being flagged as on the banned list. The gamblers are named AD and PY in the case documents. PY was listed as being banned since 2005 and he was able to gamble due to a misspelling of his name on the banned list.

AD was put on the list by his person in 2006 and was given a cash advance last year for $5,000 and was able to play blackjack for three hours before he was found at the casino. It was not until a third cash advance attempt that the player was determined to be on the list and was excluded from game play but by then it was too late.

## Rule change at two casinos in Las Vegas for Blackjack

**An apparently tiny rule change at two casinos in Las Vegas will have pretty serious negative consequences for gamblers**

**USPoker.com** An apparently tiny rule change at two casinos in Las Vegas will have pretty serious negative consequences for gamblersreported that the **Las Vegas Sands company** **just changed its payout rules for blackjack at the Venetian and Palazzo casinos in a way that greatly hurts players’ chances of coming out ahead.**

In **blackjack**, players receive two cards and then decide if they want to “hit” and get more cards, or “stand” and use the cards they already have. The goal is to get a higher score than the dealer, based on the values of the cards, without going over 21. Should you do this, you get a payout of 1-to-1; you win as much money as you bet.

A special situation happens when the first two cards dealt are a 10 and an ace (valued at 11), adding up to 21 right away, a situation called a natural blackjack. In this case, the standard payout, and the old rule at the Venetian and Palazzo, is 3-to-2. This means that if someone bets $10, they will win $15 when getting a blackjack.

**Now, at blackjack tables at the Venetian and Palazzo, the payout for a blackjack has been reduced to 6-to-5, that $10 now just wins $12 instead of $15.**

This seems like a small change, but it has a pretty serious effect on the game. Natural blackjacks are not completely uncommon; about 1 in 20 hands will come up with a natural 21. The rule change means that the casinos will be paying out quite a bit less money.

In terms of the industry, the rule change greatly increases the “house edge.” This is how casinos make their money. Games are set up to be slightly unfair to players in the long run, paying out a little bit less in total than what is taken in.

The house edge is usually expressed as a percentage. A house edge of 2% for a game means that, on average, for every $100 in bets made by players on that game, the house will pay out $98 to winners and keep $2.