Introduction to Probability

Table of Contents

In this lesson, we'll discuss the definition of probability, introduce some related terminology, and work through some example problems that require calculating the probability of certain events. Here's how I recommend working through this lesson:

1. Introduction to Probability

Return to Top

2. Concepts and Definitions Related to Probability₁

Probability

The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an infinite number of times.

The sample space, S, is the set of all possible outcomes in a setting of interest.

An event is a subset of the sample space for which we are interested in finding the probability.

Probability of Event A

$LaTeX: P(A) = \frac{\text{size of event }A}{\text{size of } S}$

Disjoint or Mutually Exclusive Outcomes

Two outcomes are called disjoint or mutually exclusive if they cannot both happen in the same trial. For instance, if we roll a die, the outcomes 1 and 2 are disjoint since they cannot both occur on a single roll. On the other hand, the outcomes 1 and “rolling an odd number” are not disjoint since both occur if the outcome of the roll is a 1. The terms disjoint and mutually exclusive are equivalent and interchangeable.

Calculating the probability of disjoint outcomes is easy. When rolling a die, the outcomes 1 and 2 are disjoint, and we compute the probability that one of these outcomes will occur by adding their separate probabilities:

$LaTeX: P(1 \text { or } 2) = P(1) + P(2) = 1/6 + 1/6 = 1/3$

What about the probability of rolling a 1, 2, 3, 4, 5, or 6? Here again, all of the outcomes are disjoint so we add the probabilities:

$LaTeX: \begin{aligned} P(1 \text{ or } 2\text{ or } 3 \text{ or } 4 \text{ or } 5 \text{ or } 6) &= P(1) + P(2) + P(3) + P(4) + P(5) + P(6) \\ &= 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 \\ &= 1 \end{aligned}$

The upcoming practice problems reference a deck of 52 cards. The image below displays the representations of the 52 unique cards in a deck. The 52 cards are split into four suits: ♣ (club), ♦ (diamond), ♥ (heart), ♠ (spade). Each suit has its 13 cards labeled: 2, 3, ..., 10, J (jack), Q (queen), K (king), and A (ace). Thus, each card is a unique combination of a suit and a label, e.g. 4♥ and J♣. The 12 cards represented by the jacks, queens, and kings are called face cards. The cards that are ♦ (diamonds) or ♥ (hearts) are colored red while the other two suits are typically colored black.

A typical deck of 52 playing cards, 4 suits, Ace through King, red (hearts & diamonds), black (clubs & spades).

Complement of an Event

Rolling a die produces a value in the set {1, 2, 3, 4, 5, 6}. This set of all possible outcomes represents the sample space (S) for rolling a die. We often use the sample space to examine the scenario where an event does not occur.

Let $LaTeX: D = \{2, 3\}$ represent the event that the outcome of a die roll is 2 or 3. Then the complement represents all outcomes in our sample space that are not in , which is denoted by $LaTeX: D ^C= {1, 4, 5, 6}$ . That is, LaTeX: D^C is the set of all possible outcomes not already included in . Figure 1 shows the relationship between D, Dc, and the sample space S.

#'s 1-6 are in region S. Values 2 & 3 are grouped & labeled as "D". Values 1, 4, 5, & 6 are grouped & labeled as D^C.

Figure 1: Event $LaTeX: D = \{2, 3\}$ and its complement, $LaTeX: D ^C= {1, 4, 5, 6}$ . LaTeX: S represents the sample space, which is the set of all possible events.

Complement

The complement of event LaTeX: A is denoted LaTeX: A^C , and represents all outcomes not in . and are mathematically related:

LaTeX: P(A) + P(A^C) =1 , i.e. LaTeX: P(A) = 1- P(A^C)

Independence

Just as variables and observations can be independent, random processes can be independent, too. Two processes are independent if knowing the outcome of one provides no useful information about the outcome of the other. For instance, flipping a coin and rolling a die are two independent processes – knowing the coin was heads does not help determine the outcome of a die roll. On the other hand, stock prices usually move up or down together, so they are not independent.

The "rolling two dice" example presented in the video above provides a basic example of two independent processes. Suppose we want to determine the probability that both rolls will result in 1. Suppose the orange die is a 1, it provides no information about the outcome of the blue die.

Multiplication Rule for Independent Processes

If LaTeX: A and LaTeX: B represent events from two different and independent processes, then the probability that both and occur can be calculated as the product of their separate probabilities:

$LaTeX: P(A \text{ and } B) = P(A) \times P(B)$

Similarly, if there are LaTeX: k events LaTeX: A_1, A_2, ..., A_k , from independent processes, then the probability they all occur is

$LaTeX: P(A_1) \times ... \times P(A_k)$

Return to Top

3. Additional Practice Problem

There are 3 questions on a True or False quiz. A student did not study for the quiz and decided to randomly guess “True” or ”False” on each question. What is the probability that the student answers exactly two of the questions correctly?

Check your answer or watch the step-by-step solution video

The probability of that the students answers exactly two of the questions correctly is 0.375.

Lesson 9 - Problem 1 Video

References

[1] The section "Concepts and Definitions Related to Probability" (C) 2024 by Erin Howard is licensed under a CC BY SA 4.0 International license. It is adapted (excerpts and some material added) from chapter 3.1 of Advanced High School Statistics (C) 2022 by David Diez, Mine Cetinkaya-Rundel, Christopher Barr, and OpenIntro licensed under a CC BY-SA 3.0 Unported license.

Student Feedback

Please view the feedback for this course on the Student Feedback - ST 243Z page.