Home > Math > Castles in the Desert: Games of Chance and Math

Castles in the Desert: Games of Chance and Math

Introduction

early-vegas

modern-vegas

Casinos represent a mind boggling concentration of resources. It is not uncommon for casinos to be vast structures with hundreds of employees, restaurants, shopping , hotels, lavish finishes, advanced technology and extensive areas for gambling activities. All of these things are concentrated in the most unlikely of places, in a desert landscape where the blazing sun is plentiful and water is sparse. Under normal circumstances populations grow around areas high in resources like places where food can grow easily or that have access to large bodies of water. The casino hubs though, especially those in arid places like Nevada, grew because the business of gambling is extremely lucrative. The business is so lucrative that these companies can even build modern day castles in the desert.

 

 

The House Always Wins

The first thing to consider when looking at the math of gambling is the idea that the house always wins. The house, or the establishment that hosts the gambling activities, does not win every bet. There are times where individuals win both small and large sums of money. What we mean by the house always wins is that in the long term, over many, many betting transactions, the house is mathematically destined to win. For each winner there are many more losers. If the house always wins in the long run, you might ask: How often does the house win and by how much?

We’ll use a few concepts from an area of math called probability to gain an idea on how to answer that question. We’ll just scratch the surface, but it will give you an idea on how to learn more.

Basic Concepts

  • Randomness
    • Things happen without an order or predictable pattern
  •  Experiment
    • An experiment is when we do a random act to get an outcome. So we might roll a dice to see which number appears.
  • Event
    • The result of an experiment. In the dice rolling example we might roll a three.
  • Sample Space
    • All of the possible outcomes or events. A cube shaped dice has six possible outcomes.
  • Probability
    • The chance something will happen. Usually expressed as a fraction. The probability of all events in a sample space adds up to 1. I have a one in six chance that a three will be rolled on a dice
  • Independent Events
    • Events are independent if they do not influence one another. Each time I roll a dice it does not matter what has been rolled before.
  • Dependent Events
    • Events are dependent if they influence one another. If I draw a card and do not replace it in the deck, then I can’t draw that card again. Specifically, if I had 10 cards and removed one, the next time I drew a card there would only be nine cards to choose from.

One Dice

Let’s think about one dice. How many ways are there to roll the dice? A cube has 6 sides so there are six ways to roll one dice.

dice2

Two Dice

What about if we roll two dice? Well there are 6 ways to roll the first dice and six ways to roll the second. The first conceptual challenge here is how do we count the events in the sample space? It is always easier to use real numbers so let’s consider that we roll 1 on the first dice. How many ways can the second dice be rolled? (6) We can repeat the thought experiment by rolling a 2 first, then a three first etc. We get something that looks like 6+6+6+6+6+6 or 6*6. We can also visualize this by organizing the outcomes in a table.

dice-combinations

We can go further and look at how many times each sum appears in the table. We can use this information to determine the probability for each sum. For example a sum of two can be formed exactly one way (by rolling two ones) in all 36 combinations. Therefore the probability of the dice adding up to 2 is 1/36. The full distribution looks like this:

dice-probabilities

 

Craps

Craps is a gambling game where participants place bets on the results of rolling two dice. The rules of the game are as follows:

  • The player (known as the shooter) rolls a pair of fair dice
    • If the sum is 7 or 11 on the first throw, the shooter wins; this event is called a natural.
    • If the sum is 2, 3, or 12 on the first throw, the shooter loses; this event is called craps.
    • If the sum is 4, 5, 6, 8, 9, or 10 on the first throw, this number becomes the shooter’s point. The shooter continues rolling the dice until either she rolls the point again (in which case she wins) or rolls a 7 (in which case she loses).

There are a wide variety of bets that relate to these activities. The purpose of this article is not to encourage gambling or explain the intricacies of the activity so we’ll only scratch the surface of this game. In fact the goal of this article is to help you understand how much a gambler can stand to lose in the long run.

Consider the case of winning on the first throw. There are 6 ways to roll a seven and two ways to roll an eleven. This means there are eight ways out of 36 to have a winning outcome.  We can pair this information with two related bets on this outcome

The seven bet pays out 4:1. So if a dollar is bet, the player nets $4 otherwise he loses his dollar. The gambler will win one in six tries in the long run or 16.7% of the time. So if the gambler places 1,000 one dollar bets he will win back $668  (167 * $4). However he will lose $833 (833 * $1) This means over the long run he walks out $165 poorer.

The eleven bet pays out 15:1. So if a dollar is bet,  the player nets $15 otherwise he loses his dollar. Similar to the 7 bet calculation , the player has a 5.6% chance of an 11 bet succeeding. If he places 1,000 one dollar bets, he will win back $840 (56 * $15) and lose $944 (944 * $1). The means over the long run he walks out $104 poorer.

In both of the previous cases the house was the long term winner. Even though the player may feel like they are winning at some points in time, over many betting cycles the house amasses a large influx of cash. The house’s winnings  are always net positive in the long run and the player’s are always net negative.

We can use the simulator in the references below to gather the long term results of all craps bet types  for our 1,000 one dollar bet scenario. Any way you go you will walk out poorer over the long run.

long-term-outcome-craps-for-the-player

Further analysis can be done on other games of chance. The outcomes are similar with the house winning in all scenarios over the long run. Some games like Black Jack provide the house a smaller edge allowing for skilled players to reap rewards. One can even observe the flow of the game and use probability to make large bets when the hands left in the deck are in the player’s favor. This is a method called card counting. This can be multiplied by having multiple players gathering information on the deck. However this practice, known as team play, is prohibited and casinos will quickly escort out or take action against players found doing it.

Just such a scenario went down when a group of MIT students devised a plan to do team play against single deck Black Jack and reap huge winnings. Their exploits are detailed in the book “Bringing Down the House”. Casinos quickly caught on to their tactics and black listed them from playing. Outside of the US the young students faced threats of physical violence when they tried to employ their tactics.

While one could argue that individuals who counted solo without the help of a team were just being good players and mastering the game, the casinos were not in the business of losing money. They quickly adjusted their games of chance to include multiple decks. This nullified the counting advantages that single decks offered. Building castles in the desert is an expensive venture and the house is going to do everything in their power to ensure that they keep winning in the long run.

So the next time you see a grand structure where there would normally not be one, think a little about why resources are concentrating there. Math is a tool that can help you solve this riddle.

Handouts

  • All Possible Outcomes of Rolling Two Dice
  • Probability Distribution of Rolling Two Dice

References

Videos

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Categories: Math Tags: , ,
  1. January 17, 2016 at 3:20 pm

    Your dedication to teaching and using your gift is amazing Charles, you poured a lot of work into this and one can tell…

  2. John
    January 28, 2016 at 3:08 pm

    Another excellent article, Charles. It reminds me of a series of lectures on arithmatic by a U of Colorado physics professor. You have an amazing mind. If only you had a friend who could … 🙂

    • January 28, 2016 at 6:17 pm

      Yes giving my professional writer friends enough notice to give feedback and listening to my editor are both things I must work on 🙂

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