The Secret Weapon of the Casino

While I personally like math, which is why I am a programmer, many people don’t like math. Math is really just a language for describing abstract things in a precise way. Predicting probability is perhaps one of the most interesting uses for math. By using a branch of math called statistics, Casino game designers are able to design games that give the house a slight edge over the player. When you add in the fact that most of the players don’t know how to calculate the odds – largely because they don’t like math – you can easily see why casinos make money. While I certainly can’t teach you statistics in a single section of an article, I can go over some of the basics so that you better understand how the casino wins.

Lets start with dice. If you take a single six sided die (I like eight and ten sided myself, but not everyone has those) and roll it, what is the chance that you will roll a three? Well, obviously the odds are one in six. There are six sides, with each side having the same chance of coming up. So, now that you have rolled a three, what are the odds of you rolling another three? If you answered one in thirty six, you are wrong! This is where probability starts to confuse people. The odds of the second roll being a three is one in six. There are six sides, with each side having the same chance of coming up. The fact that you had previously rolled a three is a fact that the die does not know, and if it did know would not care. Even if you rolled five threes in a row, the chance of the next roll being a three is still one in six! That said, the chance of a person rolling a three five times in a row is 1 in 7776.

Didn’t I just say that the odds were one in six? Well, the odds are for a single roll are one in six. The odds for any given sequence occurring are based on taking the odds of a single roll and taking that value to the power of the number of rolls in the sequence. In other words, rolling a three five times in a row is 1 in 6x6x6x6x6. Rolling a one followed by a two followed by a three followed by a four followed by a five would also be a 1 in 7776 chance. In other words, you would have to roll that die quite a bit in order to roll the 1,2,3,4,5 sequence. If you rolled long enough, you would eventually roll that sequence. That is where the casino makes it’s money. In the short term, players may win, but the longer they play the more likely it is that the casino will get the money. In fact, knowing that they will win money in the long run is why the big Vegas casinos will comp players. Giving players stuff for free may not seem like a good business strategy, but if the gifts are given as a reward for spending time gambling, then the casino will gain more than it gives away.

Having more than one die complicates things. If you have two dice and are trying to roll a three, what are the odds of at least one of the dice coming up as a three? Many of you would answer one in three. On the surface that would seem the logical result. The problem is that both the dice are independent entities. This means that of the 36 combinations, only 11 combinations have a three in them. This makes the odds 11 in 36. The missing 12th roll is actually the one occasion when both the dice are three. A nice 2.7% profit margin if you were thinking like a casino game designer.

Adding the two dice together actually forms one of the more interesting elements of statistics. The bell curve. The chance that the dice will add up to a particular value between 2 and 12 actually vary from number to number. If you were to chart this, as you can see in my chart below, you would notice that the shape forms a bell. Well, it looks more like a bell than the constellation of Libra looks like a scale. Adding more dice changes the range and amplitude, while rolling extra dice and taking the highest or lowest values will shift the bell.

					6, 1
				5, 1	5, 2	6, 2
			4, 1	4, 2	4,3	5, 3	6, 3
		3, 1	3, 2	3, 3	3, 4	4, 4	5, 4	6, 4
	2, 1	2, 2	2, 3	2, 4	2, 5	3, 5	4, 5	5, 5	6, 5
1, 1	1, 2	1, 3	1, 4	1, 5	1, 6	2, 6	3, 6	4, 6	5, 6	6, 6
2	3	4	5	6	7	8	9	10	11	12

Dice are just one element of casino games. A bigger element is cards. This is a bit more complicated then dice, but lets take a brief look. Take a deck of standard playing cards with the jokers removed. You have 52 cards. Thoroughly shuffle the cards. What are the odds of the top card being an ace of spades? If you didn’t answer 1 in 52, then I suggest you read the top part of this section again. Now, after drawing that ace of spades and laying it down in front of you, what are the odds of the next card being an ace of spades? This is where cards differ from dice. The ace of spades is no longer in the deck so your chance of drawing it is 0.Taking this into account, what if I asked you to draw any ace? Some of you are quick to say 1 in 13 and then remember that the deck isn’t full. The odds are actually only 3 in 51 or roughly 6%. How is this useful? Well, if you know how many cards are in the deck and which of the cards are useful to you, then you can quickly figure out the chance of improving your hand.

This leads to the question of chance for a particular hand being drawn. To handle this, you need to employ the techniques of permutations and combinations. A permutation is an order specific sequence. How many order specific combinations there are if I dealt out five cards? This can be calculated as follows: 52x51x50x49x48 for 311875200. Of course, almost all card games let you arrange the cards you are drawn however you want. There are 120 ways to arrange five cards (1x2x3x4x5) meaning the actual number of combinations from being dealt five cards is 2598960. We know that there are only 4 royal flushes possible, so the odds of being dealt a royal flush are 1 in 649740.

You get the general idea here. If you are interested in more detailed breakdowns of the odds, I will be having strategy guides for all of the casino games that are on the BlazingGames.com website which will explain the odds (with sections devoted to explaining how the odds were calculated).