4.2 Experiments and Probability of Events

We need to come to an understanding of probability and how it is used in statistics. This understanding will be developed over the course of this entire class. For now, keep in mind that probability is a number that is associated with how certain we are of the outcomes of an experiment or activity. While the word “experiment” might bring up a vision of a laboratory with bubbling solutions, when we talk about an experiment, we mean something different.  An experiment is a planned operation carried out under controlled conditions. Results of an experiment are not known before the experiment is conducted, so the outcome is uncertain.

A result of an experiment is called an outcome. If we make a list of all the possible outcomes for an experiment, we create the sample space. We typically are interested in a particular collection of the outcomes, so an event is any combination or subset of outcomes of an experiment.

The uppercase letter S  is used to denote the sample space. Because the sample space is a list of all the outcomes that are possible, we need a compact way to denote the list. In mathematics, we use curly brackets to list sets, this is called set notation, so the sample space for an experiment would be given by  S = {list of all outcomes}.

Upper case letters like A and B are typically used to represent events. Because an event is a collection or subset of the sample space, we will use set notation to list outcomes associated with events as well.  Any event should be described in words first and then the outcomes that make up the event should be listed as A = {list of outcomes corresponding to the event}.

Example 1: Describing a Sample Space and Events

Craps is a dice game, which uses two six-sided dice, as shown in the photograph. When each die is rolled, the top face will show any of the number of dots: 1, 2, 3, 4, 5, or 6. Together the dice can show sums from 2 to 36. The most basic roll in craps is the “come out roll.” One player will roll the dice and the sum of the dice is observed. Some players will win if a 7 or 11 shows and lose if a 2, 3, or 12 shows. Consider the experiment of rolling the dice on the come out roll and create the following related to this experiment:

1. Describe the experiment.
2. Define the sample space and list the outcomes.
3. Define event A as the winning roll and list the elements in event A.
4. Define event B as the losing roll and list the elements in event B.

Answers:

1. The experiment consists of rolling two six-sided dice and observing the sum of the dots (pips).
2. The sample space consists of all of the possible outcomes when two dice are rolled. S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}.
3. Let event A = getting a 7 or 11; A = {7, 11}.
4. Let event B = getting a 2, 3, or 12; B = {2, 3, 12}.

Combining Events

We often construct events by combining simpler events. All events are subsets of sample spaces, so we have seen how we use set notation to describe sample spaces and events. We can use set notation to combine sets as well.

• Union of Events: Sometimes we are interested in one event occurring or another event occurring. Event “A or B”  will be made up of all outcomes in A, or in B, or in both A and B at the same time. We can describe this idea by using the word “or” but it is also common to use the union symbol, [latex]\cup[/latex]. Event “A or B” means the same thing as [latex]A \cup B[/latex]. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. If we create the event [latex]A \cup B[/latex] we will list the outcomes in A, or in B, or both A and B. In this case, [latex]A \cup B[/latex] = {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are NOT listed twice. 
• Intersection of Events: Sometimes we are interested in two events occurring at the same time, so one event occurs and the other event occurs. We can describe this idea by using the word “and” but it is also common to use the intersection symbol, [latex]\cap[/latex]. Event “A and B” means the same thing as [latex]A \cap B[/latex]. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. Then [latex]A \cap B[/latex] = {4, 5}.
• Complement of an Event: Sometimes we are interested in outcomes that do not belong in an event A, so we are interested in what happens when A does not occur. The complement of event A is denoted A′ . We would read this as “A prime”. A′ consists of all outcomes that are not in A. An event and its compliment will not have outcomes in common and together would make up the entire sample space. For example, let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then A’ = {5, 6}.

We study statistics because when we take a sample, we want to use the information to infer something about the population. Sampling from a population is a random process, when we use a random sampling method. Probability concepts then form a foundational part of our understanding of statistics. Probability is the tool we use to decide if a particular sample result can be valid for a certain population. We will consider our sample results and decide how likely they would be to occur. This idea of likelihood is where we will spend some time, so you get a foundation in probability thinking.

There are situations in which the only way we can estimate a probability is to repeat the experiment many, many times and then calculate the proportion of times the event occurs. This important characteristic of probability experiments is known as the Law of Large Numbers which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.)

For example, if you flip one coin six times, you could observe the outcome: H, H, H, H, H, T. On the other hand, you could observe the outcome H, T, H, T, H, T. What happens if you continue flipping and keep track of the number of heads you observe? Try it by going to the Coin Flip App to run a simulation.

We know in the long run, a fair coin will land heads up about 50% of the time, so does our coin flipping result contract the theoretical probability? Not at all! Probability is not defined by what happens in the short term. However, if you repeatedly flip the coin, 20 times, 2000 times, 20,000 times, etc., the relative frequency of heads approaches 0.5, which is the true theoretical probability of heads occurring in the flip of a fair coin. P(head) = 0.5 means the event of observing a head in a future coin flip is equally likely to occur or not to occur. Equally likely means that each outcomes of an experiment occurs with equal probability. For example, if you toss a fair, six-sided die, each of the six faces (1, 2, 3, 4, 5, or 6) is as likely to occur as any other. If you toss a fair coin, a Head (H) and a Tail (T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

Probability of Equally-Likely Events

To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space.

[latex]\displaystyle{P}{(A)} = \frac{{\text{Number of Outcomes in A}}}{{\text{Total Number of Outcomes}}}[/latex]

It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair, or biased. Two math professors in Europe had their statistics students test the Belgian one Euro coin and discovered that in 250 trials, a head was obtained 56% of the time and a tail was obtained 44% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.

Mutually Exclusive Events

Two events are mutually exclusive if they cannot both occur at the same time. The intersection of two mutually exclusive events would have no elements in it and would be empty. Any set that is empty is defined as the empty set. For example, we have already discussed the complement of an event. Any event and its complement would not share any elements, so they would be mutually exclusive.  If we are curious about the probability of two mutually exclusive events happening, then we can add their probabilities.

Finding probabilities for events that are not mutually exclusive requires more care. If we just add the individual probabilities, then we are double counting probability where the two events share elements. Double counting occurs where the two events overlap. To correct for this issue, we can add the probabilities of the individual events as long as we subtract the probability where the intersection occurs. This brings us to several formulas for finding probabilities of combinations of events.

Probabilities of Combinations of Events

If A and B are Mutually Exclusive Events:

• [latex]P( A \cup B ) = P(A) + P(B)[/latex]
• [latex]P( A \cap B ) = 0[/latex]

If A and B are not Mutually Exclusive Events:

• [latex]P( A \cup B ) = P(A) + P(B) - P(A \cap B)[/latex]. Note: For events that are not mutually exclusive, we will count twice any outcome that is a member of both events A and B, so we must take care to subtract the probability related to the intersection of A and B.

For an event A and its Complimentary Event A’ .

• [latex]P(A) + P(A' ) = 1[/latex]
• [latex]P(A) = 1 - P(A' )[/latex]

In the previous example, the information about region and health insurance were presented in a table. A contingency table provides a way of portraying data that can facilitate calculating probabilities. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another.

Videos

This YouTube video gives more examples of basic probabilities: Introduction to Probability

This YouTube video gives examples of probabilities involving AND.

This YouTube video gives examples of probabilities involving OR.

This YouTube video gives examples of finding probabilities related to mutually exclusive events.

This YouTube video gives an example of finding probabilities using a contingency table.

Sources

Example 4 was used from Statway College Module 2, by Carnegie Math Pathways is licensed under CC BY NC 4.0.

¹NHTSA “Distracted Driving”. Available online at https://www.nhtsa.gov/risky-driving/distracted-driving (accessed 3/14/20240