Probability and Games: What is Luck?
Chances are you’ve heard someone you know question probability in a game setting; maybe without even knowing it. Whether the throw of a single die, in a game of pure odds like flipping a coin or roulette, or even a game of multiple dice like Risk or Dungeons and Dragons, someone has claimed a certain outcome ‘must’ be coming next, or that some other outcome was ‘unfair’.
The basic principles of probability can be relatively simple to understand, but the applications to competitive games or scenarios where outcomes matter can be difficult – not because we can’t understand them, but because we often have a vested interest in the outcome. In this brief article, I’m going to talk a bit about what underpins probability in games, and then give a few examples of these theories in practice – and hopefully answer some questions at the same time.
A core principle of probability theory in general and how it applies to games in particular, is that of ‘independence’. In basic terms, independence exists when one event in a set does not hold any effect over the outcome of the other events. For example, if you were to roll one die twice, the first roll has no influence over the outcome of the second roll, and is therefore independent.
However, assume that you are a contestant with a chance of winning one of three prizes: prize 1, 2 and 3; each contestant in the room has a raffle ticket which could be matched to a prize. Also assume that each prize is assigned as the winner is called, and the ticket discarded after the draw. Even if tickets are pulled randomly, if the winning ticket for prize 2 is pulled and the winning ticket discarded, the odds of winning prizes 1 and 3 have been effected. The second draw is therefore dependent on the result of the first draw, and the two events are not independent.
In games, the sample size refers to the number of times you have repeated a probability-based event: flipped a coin, rolled a die, etcetera. Here, we are not using a sample to predict or measure an outcome, but rather as the number of events increases, the closer the results will be to an expected outcome. This is why, for example, if you flip a coin 15 times, you may end up with 10 ‘tails’; but, the more times you flip that coin the closer your results come to 50% ‘heads’, 50% ‘tails’. It also helps to explain lucky or unlucky streaks in dice-based games or probability based casino games: in the scope of a single game, your sample size is small.
Much of the work that we publish via the Abacus Insider or on Abacusdata.ca are based on online surveys, and in those releases, we discuss the sampling methodology used to generate the results. This sort of sample discussion is a bit different than rolling dice. In these cases, the goal of sample size is to create a ‘representative sample’, or one that proportionately mirrors the whole population we are trying to measure, whether that is national, provincial, or even city level. By using weighting and balancing, we are able to produce a group of respondents which accurately represents the overall population, and thus allows us to generate accurate data within a margin of error. For the purposes of this article, think of sample size as strictly the total number of events, whether it be rolls, flips, spins, or what have you.
How about some specific examples?
Here I’m going to briefly answer some common questions about probability in games. The aim is not to go into the detailed odds of specific outcomes, but rather to deal with common misconceptions or misunderstandings about probability in practice.
Roulette: The wheel came up black the last five times; it must be red this time…
This is, perhaps, the most commonly held misconception in casino gaming. In fact, the existence of this misconception is the reason casinos have a board above their roulette tables detailing the history of the last several spins. For the purpose of this example, we will assume that the single green space doesn’t exist and that the odds are truly 50/50, rather than ~49/49.
Think back to independence. Each spin of the wheel is an independent event: one spin (on a fair wheel) cannot influence the results of the second spin. Thus, the odds of each spin coming up red (R), or black (B), is 1 in 2. You might ask: “but isn’t it really rare for there to be five blacks in a row”? You would be right, if you were trying to predict the next five spins at once, but you aren’t – you are only guessing at the immediate next spin. The odds of predicting five consecutive spins are (1/2)^5, or 1 in 32. Actually, those are the odds of trying to predict any series of five consecutive outcomes of a 50/50 event, wither it is RBBRB, BBRRB, BBBBB, RRRRR, etecera.
Dice Games: Was that really a lucky streak of rolls?
A huge range of games use dice to introduce some degree of randomness. Dice in Monopoly dictate number of spaces, in Risk they decide combat, and in Dungeons & Dragons they decide everything from character attributes to specific chance-based scenarios. However, randomness in each of these cases are influenced by a range of different factors which, if understood, help to remove the mystery behind the rolls.
Online you can find lengthy discussions and analyses of the specific odds of these and many other dice-based games, so I’m not going to address those sorts of detail here. I will however, discuss the differences in probability factors at play across a few types of dice games.
A common misconception with games that use two six-sided dice is that each roll has an equal chance of resulting in each number from 2-12. In most of these games, the face value of the two rolls are added together to produce a total value; say, how many squares you have to move. Therefore, the odds of rolling a number depend on how many ways that number can be reached by adding values from one to six. For example, there is only one way to roll a two, but multiple ways to roll a seven. In fact, there is a 2.8% chance of rolling a two or a twelve, while there’s a 16.7% chance of rolling a seven. Make these rolls enough times (sample size), plot them in a bar graph, and you’ll start to see a ‘normal distribution’ of results.
Some games introduce more complex mechanics which impact the odds as they relate to gameplay. In Risk for example, defending players roll with up to two dice and win ties, while attacking players roll with up to three dice. Although these rules have gameplay consequences, they do not affect the overall odds related to rolling dice – the same theories apply.
Because these games don’t usually involve enough rolls to create a sample size sufficient which would allow us to observe an obviously normal distribution, players often invoke lucky rolls, lucky dice, or a lucky method of throwing the dice – but it’s all just probability.
Although probabilities can seem complex, the concept itself is relatively straightforward. However, as humans, our minds can play perceptive games, making us think we see patterns where we shouldn’t, or missing patterns where they should be obvious.
Probability happens to be an empirical form of these sometimes deceptive patterns or observations, but they are ubiquitous, (personal testimonials, or eyewitness testimony, for example). If we think more carefully about what we are actually observing, and look at what empirical evidence can show us, the mystery often becomes clearer.
So, the next time you’re at the roulette table and your friend tells you to bet black because it’s been red the last five times in a row, send them the link to this page.