Chapter 3 Probability and random events

Probability is a method of mathematically modeling a random process so that we can understand it and/or make predictions about its future results. Probability is an essential tool for casinos, as well as for banks, insurance companies, and any other businesses that manage risks.

Chapter goals

In this chapter we will learn how to:

  • Model random events using the tools of probability
  • Calculate and interpret marginal, joint, and conditional probabilities
  • Interpret and use the assumptions of independence and equal outcome probability

This chapter uses mathematical notation and terminology that you have seen before but may need to review. If you have difficulty with the math, please refer to the sections on Sets and on Functions in the Math Review appendix.

Example application: Roulette

We will develop ideas by considering the casino game of Roulette. The picture below shows what a roulette wheel looks like.

roulette wheel image Source: Roulette Vectors by Vecteezy

Here are the rules:

  • It features
    • a ball.
    • a spinning wheel with numbered/colored slots.
    • a table on which to place bets
  • The slots are numbered from 0 to 36
    • Slot number 0 is green
    • 18 slots are red
    • 18 slots are black.
    • The picture above depicts an American roulette table, which has an additional green slot labeled “00”,
    • I will assume we have a European roulette table, which does not include the “00” slot.
  • Players can place various bets on the table including:
    • Red (ball lands in a red slot) pays $1 per $1 bet
    • Black (ball lands in a black slot) pays $1 per $1 bet
    • A straight bet on any specific number (ball lands on that number) pays $35 per $1 bet

Like other casino games, a roulette game is an example of a random process. Something will happen, it matters (to the players and the casino) what will happen, but we don’t know in advance what will happen.

3.1 Outcomes and events

To build a probabilistic model of a random process, we start by defining the outcome we are interested in. An outcome can be a simple yes/no result, it can be a number, or it can be a much more complex object. The outcome should be a complete description of the random process, in the sense that everything we are interested in can be defined in terms of the outcome.

Outcomes in roulette

The outcome of a single game of roulette can be defined as the number of the slot in which the ball lands. Call that number \(b\).

The set of all possible outcomes is called the sample space

The sample space in roulette

The sample space for a game of roulette can be defined as the set of all numbers the ball can land on: \[\Omega = \{0,1,2,\ldots,36\}\] This sample space has \(|\Omega| = 37\) elements.

Next, we define a set of events that we are interested in. We can think of an event as either:

  • A statement that is either true or false OR
  • A subset of the sample space

These two concepts are equivalent, though the subset concept makes the math clearer.

Events in roulette

These roulette events are well-defined for our sample space:

  • Ball lands on 14: \[b \in \{14\}\]
  • Ball lands on red: \[b \in Red = \{1,3,5,7,9,12,14,16,18,19,21,23,25,27,30,32,34,36\}\]
  • Ball lands on black: \[b \in Black = \{2,4,6,8,10,11,13,15,17,20,22,24,26,28,29,31,33,35\}\]
  • Ball lands on one of the first 12 numbers: \[b \in First12 = \{1,2,3,4,5,6,7,8,9,10,11,12\}\]

We could define many more events, depending on what bets we are interested in.

Since events are sets, we can use the terminology and mathematical tools for sets.

Relationships among events

In our roulette example:

  • Events are identical if they contain exactly the same outcomes:
    • The event “ball lands on 14” and “a bet on 14 wins” are identical since \(\{14\} = \{14\}\).
    • Intuitively, identical means they are just two different ways of describing the same event.
  • An event implies another event if all of its outcomes are also in the implied event
    • The event “ball lands on 14” implies the event “ball lands on red” since \(\{14\} \subset Red\).
    • When an event happens, any event it implies also happens.
  • Events are disjoint if they share no outcomes:
    • The events “ball lands on red” and “ball lands on black” are disjoint since \(Red \cap Black = \emptyset\).
    • If two events are disjoint, they cannot both happen.
    • But they can both fail to happen. For example, if the ball lands in the green zero slot (\(b = 0\)), neither red nor black wins.
  • Any two outcomes are either identical or disjoint
    • The events “ball lands on 14” and “ball lands on 25” are disjoint since \(\{14\} \cap \{25\} = \emptyset\).

3.2 Probabilities

Our final step is to define a probability distribution for this random process, which is a function that assigns a number to each possible event. The number is called the event’s probability.

Probabilties are normally between zero and one:

  • If an event has probability zero, it definitely will not happen
  • If an event has probability strictly between zero and one, it might happen.
  • If an event has probability one, it definitely will happen.

3.2.1 The axioms of probability

All valid probability distributions must obey the following three conditions:

  1. Probabilities are never negative: \[\Pr(A) \geq 0\]
  2. One of the outcomes will definitely happen: \[\Pr(\Omega) = 1\]
  3. For any two disjoint events \(A\) and \(B\), the probability that \(A\) or \(B\) happen is the sum of their individual probabilities: \[\Pr(A \cup B) = \Pr(A) + \Pr(B)\]

These conditions are sometimes called the axioms of probability. Probability distributions have many other properties, but they can all be derived from these three.

Outcome probabilities for a fair roulette game

Let’s assume that the roulette wheel is “fair” in the sense that each outcome has the same probability. Now, I should emphasize that this doesn’t have to be the case, it’s just an assumption. But it’s a reasonable one in this case because casinos are required by law to run fair roulette wheels and would be subject to heavy penalties if they run unfair wheels. Later on, we will use statistics to confirm that a roulette wheel is fair.

Call that probability \(p\): \[p = \Pr(b = 0) = \Pr(b = 1) = \cdots = \Pr(b = 36)\] To find the value of \(p\) we use the rules of probablity. By rule #2 of probability, one of the outcomes will happen: \[\Pr(\Omega) = 1\] Since the different outcomes are disjoint, rule #3 implies that: \[\underbrace{\Pr(\Omega))}_{1} = \underbrace{\Pr(\{0\})}_{p} + \underbrace{\Pr(\{1\})}_{p} + \cdots + \underbrace{\Pr(\{36\})}_{p}\] Summarizing this equation: \[1 = 37p\] Solving for \(p\) we get: \[p = 1/37 \approx 0.027\] That is, each of the 37 outcomes have a probability of \(1/37\).

Since this is an introductory course, our sample space will usually contain a finite number of outcomes, as in our roulette example. In that case, probability calculations are pretty simple:

  • Find the probability of each outcome.
  • To find the probability of a specific event, just add up the probabilities of its outcomes.

Event probabilities for a fair roulette game

In the roulette example, the probability of any event \(A\) is just the number of outcomes in \(A\) times the probability of each outcome \(1/37\): \[ \Pr(A) = |A|*1/37\] The notation \(|A|\) just means the size of (number of elements in) the set \(A\).

For example: \[\Pr(b=25) = |\{25\}|*1/37 = 1/37 \approx 0.027\] \[\Pr(Red) = |Red|*1/37 = 18/37 \approx 0.486\] \[\Pr(Even) = |Even|*1/37 = 18/37 \approx 0.486\] \[\Pr(First12) = |First12|*1/37 = 12/37 \approx 0.324\]

However, not all sample spaces contain a finite number of outcomes. For example, suppose we are interested in using probability to model the unemployment rate, or a person’s income. Those are real numbers, and can take on any of an infinite number of values. This adds a few complications, and is the reason that the probability axioms refer to events (sets of outcomes) and not individual outcomes.

What do probabilities really mean?

What does it really mean to say that the probability of the ball landing in a red slot is about 0.486? That’s actually a tough question. There are two standard interpretations for probabilities:

  • Frequentist or classical interpretation: we are thinking of the random process as something that could be repeated many times, and the probability of an event is the approximate fraction of times that the event will occur. That is, if you go to a casino and bet 1000 times on Red, you will win about 486 times.
  • Bayesian or subjectivist interpretation: the random process is a one-time occurence, but we have limited information about it and the probability of event represents the strength of our belief that the event will happen.

The frequentist interpretation of probability is well-suited for simple repeated settings like casino games or car insurance, while the Bayesian interpretation makes more sense for things like predicting election results.

  1. You may wonder: if it makes more sense to describe independence in terms of conditional probabilities, why do we define it in terms of joint probabilities? The key is the requirement that the events have nonzero probability. When \(B\) has zero probability the conditional probability \(\Pr(A|B)\) is not well defined since its denominator is zero.↩︎

  2. If you are interested in learning more about this, an article in Science provides an overview of the controversy, and a blog post by statistician Andrew Gelman provides a thourough discussion of the statistical issues.↩︎