Lucky numbers

The winning strategies for casino blackjack players are based on the knowledge of the changing composition of the deck. Actually, every card has a point value. The point value is considered positive if the card out of the deck favors the player and negative if it favors the casino. The magnitude of the point value reflects the magnitude of the effect of the cards but for practical purposes it is usually chosen to be a small integer. Then the cumulative point count is taken to be proportional to the expectations of the player. The player’s corresponding expectations and best strategy depend just on the fractions of each type of card at present in the pack and changes slowly only with the pack’s size. In that way the system normalize by dividing the cumulative point count to the total number of unseen cards at this time. Most of the point count systems are initialized at 0 cumulative total for the total pack and the normalized cumulative count is taken to indicate the change in player’s expectation from the value for the full pack. The idea that stays behind these point count systems is to assign to each card point values which are proportional to the observed effects of deleting a “small quantity” of that card. Players have to compromise between accuracy and simplicity (small integer values). The “ultimate strategy” is a point count based on moderate integer values which fits quite closely the data available in the early 60’s.

Some people are familiar with Julian Braun’s simulation of different point count systems. “Bet 1 to 4” means that 1 unit was bet except for the most advantageous x % of the situations, when 4 was bet. In order to compare systems, x was taken to be approximately the same in every case (21 %).

1. Basic Braun + - Strategy/System: the flat bet is .2% and the bet 1 to 4 is 1.4%;
2. Braun + - Strategy/System: the flat bet is .7% and the bet 1 to 4 is 2.0%;
3. Revere Pt. Ct. Strategy/System: the flat bet is .6% and the bet 1 to 4 is 2.1%;
4. Revere Adv. + - Strategy/System: the flat bet is .5% and the bet 1 to 4 is 1.6% to 1.8%+;
5. Revere Adv. Pt. Ct. -71 Strategy/System: the flat bet is .6% and the bet 1 to 4 is 2.0%;
6. Revere Adv. Pt. Ct. -73 Strategy/System: the flat bet is .8% and the bet 1 to 4 is 2.1% to 2.3%+;
7. Thorp Ten Count Strategy/System: the flat bet is .7% and the bet 1 to 4 is 1.9%;
8. Hi-Opt Strategy/System: the flat bet is .8% and the bet 1 to 4 is 2.1% to 2.3%+;

Twenty-one or blackjack is a card game that is played all over the world. In the beginning of the game, the dealer shuffles the cards and the bets have to be placed. The number of decks in blackjack is usually 1, 2, 4, 6 or 8 and there is a minimum and maximum allowed bet. After the bets are placed from the players, the hands are dealt. In order to improve the hand, each of the players uses a certain strategy and skill. According to a fix strategy in the end the dealer plays out his hand. There is no skill allowed in his case and all bets are settled.
There is a well known strategy in case the play begins with randomly shuffled deck. It was proposed by Baldwin, Cantey, Maisel, and McDermott in 1956. Players who follow the strategy of the Baldwin group have the tiny edge of +.10%. The best strategy against a complete deck is known as “basic strategy”. It is approximately identical with the Baldwin group strategy and it gives to the player an edge of +.13 against 1 deck and -.53 against 4 decks.

When we toss a coin, there is a 50% chance for a tail and 50% chance for a head. If we assume that, we have the opportunity to bet that the next flip will be tail and the payoff will be $1. So, based on that example the mathematical expectation is:

(.5)(1)+ (.5)(-1)=0

For each game the mathematical expectation of any bet can be calculated by multiplying each probable loss and gain by the possibility of that loss and gain. After that we have to add the two digits.