5. Craps Math
Craps is an interesting exercise in probability because it’s a great example of a bell curve. That’s when some results happen so seldom that the drawing of the curve is low on either end, but the odds of the results in the middle happening are much higher.
Here are the possible outcomes when rolling a pair of dice:
2 – 1 +1 – Only one possible way of getting this total.
3 – 2+1 or 1+2 – Only 2 possible ways of getting this total.
4 – 3+1, 2+2, or 1+3 – Only 3 possible ways of getting this total.
5 – 4+1, 3+2, 2+3, or 1+4 – Only 4 possible ways of getting this total.
6- 5+1, 4+2, 3+3, 2+4, 1+5 – Only 5 possible ways of getting this total.
7 – 6+1, 5+2, 4+3, 3+4, 2+5, 1+6 – Only 6 possible ways of getting this total.
8 – 6+2, 5+3, 4+4. 3+5, 2+6 – Only 5 possible ways of getting this total.
9 – 6+3, 5+4, 4+5, or 3+6 – Only 4 possible ways of getting this total.
10 – 6+4, 5+5, or 4+6 – Only 3 possible ways of getting this total.
11 – 6+5 or 5 +6 – Only 2 possible ways of getting this total.
12 – 6+6 – Only one possible way of getting this total.
You only have 11 possible totals, but you have a total of 36 different outcomes.
Knowing this, you can divide the number of ways of achieving each total by 36 in order to determine the probability of getting that total.
So getting a total of 2 or 12 has a probability of 1/36.
3 or 11 has a probability of 2/36, or 1/18.
4 or 10 has a probability of 3/36, or 1/12.
5 or 9 has a probability of 4/36 or 1/9.
6 or 8 has a probability of 5/36.
7 has a probability of 6/36, or 1/6.
So your most likely outcome is a total of 7, but that still only happens 1 time out of 6.
But you can bet on any of these totals at various times in the game. You can compare the payoffs on these bets with the odds of winning to determine the house edge on each of those bets.
For example, you can make a place bet on any 8 or any 6 and get a payoff of 7 to 6 if you win. But the odds of winning that bet are 5/36. That can be converted into a percentage, and we can calculate the house edge for that bet. The odds of winning this bet are 13.89%.
Place this bet 100 times, and you will win 13.88 bets with winnings of $1.17 each time (7 to 6). That’s $16.24 in winnings. But you lose 86.12 times, losing $1 each time, for losses of $86.12. You’ve lost way much more than you’ve won over these 100 bets–$69.88. That makes the house edge 6.99% on this bet, which is almost 7%. That’s worse than roulette with its 5.26% edge.
Luckily, many of the bets on the craps table have a much lower house edge.
Blackjack is the subject of my preferred gambling math. This classy game is also one of the few in the casino where a good player might get an advantage. The game has a memory, which makes it more intriguing.
This is what I mean:
On each turn of the roulette wheel, the chances are the same. The probabilities of what will happen on the next spin are unaffected by the results of previous spins. Each time the wheel is spun, there are 38 potential outcomes, and each one has an equal chance of occurring.
The chances would vary with each spin, though, if you removed a slot from the wheel after it was hit.
Here is an illustration:
You place a bet on black. That wager has an 18/38 chance of success.
You succeed. The ball is kept in that slot by the croupier (the roulette dealer), preventing further landings on that spot.
You place a new wager on black. Since one of the possibilities has been eliminated, the chance of winning this time is merely 17/38.
Every time a card is dealt in blackjack, this is exactly what occurs. In the following rounds, only 51 of the 52 possibilities are still accessible.
Until the dealer deals another round of cards, this continues.
It goes without saying that the chances are always the same in a game with a continuous shuffler.
However, the majority of games are still dealt by hand without the aid of a machine. In these games, you may roughly keep track of the cards dealt and increase your bets when your chances of earning more money are better.
That works like this:
The payout for a "natural," sometimes known as "blackjack," is 3 to 2. A two-card hand with a 21-point total is that. Aces, which count as 11, and the 10, J, Q, and K, each of which counts as 10, are the only card values that can produce such a hand. All other card values, such as J, Q, and K, do not.
Blackjacks are impossible to obtain if a deck's aces are all gone. You simply cannot accomplish this.
The likelihood of getting a blackjack also decreases each time a 10 is handed.
However, the chances shift a little bit in favor of the player each time a lower-ranked card, such as a 2, 3, 4, 5, or 6, is dealt.
Because of this, a card counter will employ a system to roughly track the ratio of high cards to low cards. High cards are given a score of 1, whereas poor cards are given a score of 1. The counter is aware that he has a higher than usual probability of receiving the 3 to 2 payoff if and when the count increases significantly on the positive side. As a result, he increases his wagers. More wagers are placed by him as the count rises.
As soon as the count is zero or negative, he reduces his wager.
But those are the fundamentals of card counting. There is much more to it. They also have a mathematical foundation.