The Adding Rule

(a.k.a. “The fundamental principle of counting”)

If you have two processes, and one of them can be done in \(x\) ways, while the other process can be done in \(y\) ways, then the total number of ways you can do one of the two processes would be the sum \(x+y\).

This is obvious, as you will see: If flipping a coin gives you two possible outcomes (heads or tails), and rolling a die can give you six (numbers one to six), then there is a total of \(6+2=8\) different results from the “flip a coin or roll a die” process (heads, tails, and the numbers one to six).

Mutually Exclusive Processes

Caution: I will now introduce a fancy term for something simple.

If there are two mutually exclusive processes, \(X\) and \(Y\), with \(x\) and \(y\) outcomes respectively, then there are \(x+y\) different outcomes in total for the process “do \(X\) or \(Y\)”.

Two processes are called mutually exclusive if they do not share a common outcome. This depends on what you take as an outcome:

• You can call the number on the top face of a die as an outcome e.g. five, or three
• You can also call the position where the die had landed as an outcome e.g. it landed here, or on top of the black square, etc.
• How you rolled the die, e.g. throwing or shaking it, can also be considered as the outcomes.

Fliping a coin and rolling a die are mutually exclusive, as we consider the outcomes of rolling a die as different from heads or tails.