Probability is the likelihood of an outcome. Before we can properly define probability, we must first define 'events.' It is helpful and convenient to denote the collection of events as a single letter rather than list all possible outcomes.

Often students in introductory statistics courses struggle with probability due to getting caught up and/or confused with the general notation used in describing events and the associated probability. To put it simply, the notation is shorthand to keep one from continually needing to write out long phrases to explain what is taking place. For instance, if one were to consider the toss of a fair coin the common theme is that there is a 1/2 chance, or 0.5 probability, of the coin coming up Tails.  But how does one write this event?

Converting to Probability Notation

1. Identifying the outcome event of interest: {Getting a Tail when we toss a fair coin}.
2. Use a single letter or word to represent this outcome of interest: T={Getting a Tail when we toss a fair coin}, for instance.
3. State your interest in the probability of this outcome: P(T) which is read, "Probability of getting a Tail when we toss a fair coin."

When you read a statistics text book a common lettering system uses the beginning of the alphabet. That is, the authors use 'A', 'B', etc. to define outcome events of interest.

As you can see, the lettering can become convoluted! Just remember that the key is to identify what your outcome event of interest is.

When you first start learning about sets and events, it is often helpful to consider the outcome space.

Now that we know how to denote events, the next step is to use the set notation to represent set operations. Each operation will also be presented in a Venn Diagram. Set operations are important because they allow us to create a new event by manipulation of other events

Before discussing the specific rules of probability, it's important to recognize there are three main interpretations of probability.

This section will provide the basic terms and properties associated with classical probability. We generally focus on classical probability but the probability properties apply to classical and subjective probabilities.

The concept of conditional events and independent events determines whether or not one of the events has an effect on the probability of the other event. You can possibly imagine several daily conversations you may have that invoke these concepts. For instance, say you are discussing driving directions with a friend on the quickest way to get to some destination. The friend asks if leaving during rush hour would change these plans. If you respond that it does, this reflects the concept of "conditional" as the quickest way is conditioned, or affected, by the time of day your friend leaves. If you respond that it does not, this reflects the concept of "independence" as the quickest way is the same regardless of when your friend leaves. The time of day does not affect the quickest way.

Another example is practically any sporting event. For instance, a team might have a probability of 0.6 of winning the Super Bowl or a country a probability of 0.3 of winning the World Cup.

• Conditional Scenario: What if it rains the team's chances may change (for the better or possibly for the worse)? The probability of winning is affected by the weather - conditional. 
• Independent Scenario: What if the game is being played in an enclosed stadium? In such a case the weather may have no effect on the team's chances - independent.

The two-way table below displays the World Campus enrollment from Fall 2015 in terms of level (undergraduate and graduate) and biological sex.

Choose one student from the sample, what is the probability that the student is a female?

\(P(F)=\dfrac{6027}{12242}\)

If it is known that the student is a graduate student, what is the probability that the student is a female?

We now are interested in only graduate students who are female (2213). The total is just the number of graduate students (5000).

\(P(F|G)=\dfrac{2213}{5000}\)

Note that this is not the same as the probability that a selected student is a female graduate student (ie. \(P(F \cap G\))).

In this scenario, the marginal probability is not the same as the conditional probability. This means that given the student is a graduate, changes the likelihood that the student is a female. This is reflective of events that are not independent i.e. they are considered dependent events. Independent events will be discussed in more detail later in the lesson.

The Venn diagrams below will help you visualize why this is true.

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To introduce the idea of independence let's look at a table similar to the one on the previous page. The two-way table below displays the World Campus enrollment predictions for a future semester in terms of level (undergraduate and graduate) and biological sex.

Choose one student from this future semester, what is the probability that the student is a female?

\(P(F)=\dfrac{6600}{12000}=.55\)

If it is known that the student is a graduate student, what is the probability that the student is a female?

We now are interested in only graduate students who are female (2750). The total is just the number of graduate students (5000).

\(P(F|G)=\dfrac{2750}{5000}=.55\)

In this example, the marginal probability is the same as the conditional probability. This will be reflective of events that are independent. In other words knowing the selected student is a graduate student does not give us any additional information about the gender of the student.

We can see that in some situations, the conditional and marginal probabilities are different and for some situations, the two probabilities may be equal. When they are equal, the events are independent events, when they are not equal, they are not independent events.

Unless one is explicitly told that events are independent, one cannot simply assume that they are. However, there are some events for which we naturally assume such independence exists (e.g. a flip of a fair coin or the roll of a fair die). For these examples, one expects that the probability of an outcome of one trial does not affect the probability of subsequent trial outcomes. That is, the probability a tail comes up on the first flip of a coin does not change the probability that a tail comes up on the second, third, fourth, etc. flip of that coin.

We can check for independence of two events by showing that any ONE of the following is true. For any given probabilities for events A and B, the events are independent if:

1. \(P(A \cap B)=P(A)\cdot P(B)\)
2. \(P(A|B)=P(A)\) or,
3. \(P(B|A)=P(B)\)

Students often confuse independent events and mutually exclusive events. The two terms mean very different things. Recall that if two events are mutually exclusive, they have no elements in common and thus cannot both happen at the same time. In fact, mutually exclusive events are dependent. If A and B are mutually exclusive events, then...

\[P(A \cap B)=0 \neq P(A)P(B)\]

The above example illustrates the use of Bayes' theorem to find "reverse" conditional probabilities.

In diagnostic testing (e.g. drug tests), there are five key concepts:

Probability is important because it tells us the likelihood of events. Understanding conditional probabilities is also important because p-values, which we'll explore in furture lessons, are conditional probabilities. Now that we understand the probability of events, we can move on to understanding random variables and probability distributions, and thus more deeply into inference