# Rules of Probability

Most all statistics classes discuss probability, even if only implicitly.  For this reason, it is important to understand the rules of probability, which may be difficult for those without prior statistics experience.  I made the page below to help with this topic.  If you have any questions or comments, please email me at MHoward@SouthAlabama.edu.  I am always happy to chat about stats!

First, we need to define a few terms.  When discussing probability, an experiment is a process that leads to one of several possible outcomes.  This may be flipping a coin, rolling a dice, or something more complex (i.e. conducting a science experiment).  Each of these outcomes is called an event.  When flipping a coin, we could consider obtaining a heads as Event A, whereas obtaining a tails could be Event B.  Each possible result is a separate event.  Lastly, the sample space is all possible outcomes of the experiment.  This is a collection of all possible events, and the total probability of the sample space is 1 (100%).

We can visualize these terms as the figure below: In this figure, the entire area of the rectangle represents the sample space.  This is all possible outcomes of the experiment, which should add up to a probability of 1 (100%).  The circle represents the probability of Event A.  Because it is smaller than the sample space, its probability is less than 1 (<100%).  So, there is a chance that Event A can occur, but there is also a chance that it may not occur.  But how can we specify the chance that it cannot occur?  We can use the complement rule.

The complement rule specifies that the probability of an event NOT occurring is one minus the probability of the event occurring.  In other words, it is the Sample Space minus Event A.  Let’s look at that figure again to visualize this rule: Visually, the complement rule simply specifies that the probability of Event A NOT occurring is the space NOT taken up by Event A.  So, in the figure above, it is simply the area in the sample space not covered by Event A.

If we assume that Event A has a 25% chance of occurring, for example, then the probability of Event A NOT occurring is 1 – .25 (.75).  Therefore, we would believe that Event A has a 75% chance of not occurring.

Easy enough, right?  Well, now let’s think about two events in the sample space, Event A and Event B.  What is the probability that BOTH Event A and Event B occur?  This can be determined by the multiplication rule.

The multiplication rule specifies that the probability of both Event A and Event B occurring is the probability of Event A multiplied by the probability that Event B occurs.  In other words, it is Event A times Event B.  Let’s look at the figure below to visualize this rule: When we are interested in BOTH Event A and Event B occurring, we are only interested in the area of overlap highlighted above.  So, we are NOT interested in all of Event A or all of Event B, but only the highlighted part of each.  To identify the size of this space, we simply multiply the probability of Event A and Event B together.

If we assume that Event A has a 25% chance of occurring and Event B has a 50% chance of occurring, for example, then the probability of BOTH Event A and Event B occurring is .25 * .50 (.125).  Therefore, we would believe that the probability of both Event A and Event B occurring is 12.5%.

NOTE: The example above assumes that the two events are independent.

But what happens when the events are mutually exclusive?  For example, flipping a single coin and obtaining BOTH a heads and a tails?  This can be visually represented by the figure below: As can be seen, there is no overlap in the events.  So, the probability of BOTH occurring is 0 when the events are mutually exclusive.

Lastly, let’s discuss how to find the probability of Event A AND/OR Event B occurring.  In other words, we want to know the probability of Event A, Event B, or Event A and B occurring.  This can be determined by the addition rule.

The addition rule specifies that the probability that Event A and/or Event B occurs is the probability of Event A plus the probability of Event B minus the probability of both Event A and Event B occurring.  Let’s start with mutually exclusive events, visualized by the figure below: As seen in the figure, we are interested in the total space covered by Event A and Event B.  Because there is no overlap, the probability of BOTH occurring is 0.  Thus, when the events are mutually exclusive, the probability of either occurring is simply the probability of Event A plus the probability of Event B.

If we assume that Event A has a 25% chance of occurring and Event B has a 50% chance of occurring, for example, then the probability of either Event A or Event B occurring is .25 + .50 (.75).  Therefore, we would believe that the probability of either Event A and Event B occurring is 75%.

But what happens when the events are not mutually exclusive?  Let’s look at the figure below: Again, we are interested in the total space covered by Event A and Event B.  As we found out when discussing the multiplication rule, the probability of BOTH Event A and Event B occurring is Event A times Event B.  Therefore, to find the probability that Event A and/or Event B occurs, we add Event A and Event B, then subtract Event A times Event B.

Some people may be confused, however, as to why we subtract Event A times Event B.  This is because we already double-counted the overlap when adding Event A and Event B, and we need to correct this double-counting.  That is, when determining the total space covered by Event A times Event B, we added the two.  This counted the overlap area with Event A as well as Event B, when it should have been only counted once.  For this reason, we must subtract the overlap area once to correct this issue.

If we assume that Event A has a 25% chance of occurring and Event B has a 50% chance of occurring, for example, then the probability of Event A and/or Event B occurring is .25 + .50 – .125 (.625).  Therefore, we would believe that the probability of Event A and/or Event B occurring is 62.5%.

NOTE: The example above assumes that the two events are independent.

That is all for probability rules.  If you have any questions or comments, feel free to email me at Mhoward@SouthAlabama.edu.  I’ll try to reply ASAP.