While the fundamental counting principle, permutations, and combinations deal with all possible combinations that can occur, probability deals with the likelihood of a specific event occurring. For example, you know that when you toss a penny it is just as likely the penny will land heads up as it will land tails up. The penny has two sides, and each side is considered one part of the penny, or one‐half. Since both sides of the penny are one‐half of the penny, the probability of the penny landing heads up is the same as landing tails up.



When you have two or more events occurring, you will perform a specific action depending on the nature of the events. Independent events mean that the outcome of two or more events has no impact on the outcomes of the other events. You find the product of the probabilities to reach the final probability.


Dependent events mean that one event changes the outcome of another event. For example, removing a marble from a bag of marbles reduces the total amount of marbles in the bag. If you start out with 20 marbles in a bag and remove one you are left with 19 marbles. However, if you remove the marble and replace it back into the bag, then the first event has no impact on the second event and would be considered independent events.


Mutually exclusive events are similar to independent events in that the outcome of one event has no impact on the outcome of the second event. The difference is that mutually exclusive events refer to a single occurrence, while independent events refer to several occurrences. A safe indicator for independent events is the word “and” placed between events, while a safe indicator for mutually exclusive events is the word “or” placed between events.

For example, in a single toss of six‐sided die numbered 1 to 6 I can roll a 1, 2, 3, 4, 5, or a 6. The likelihood of rolling one of those numbers is one‐sixth. The likelihood of rolling a 1 or a 2 is greater than the probability of rolling just one of the numbers. We add the probabilities together to reach the final probability of mutually exclusive events.