What Is And How Does The Fundamental Counting Rule Principle Work In Probability Statistics Math

In this video we discuss the fundamental counting rule or principle, we go over, through examples how the fundamental counting rule works, and how and when to use it.

Transcript/notes
Fundamental counting rule principle
Lets say that you go shopping and you are looking for some new threads. You want to get a shirt, shorts and some shoes. And what do you know, your favorite store just happens to have a sale going on. Get a shirt, shorts and shoes for $49.99.

They have the shirt you want in 7 different colors, the shorts you like in 4 different colors and the shoes in 5 different colors. And because you are a stats nerd like me, you want to know how many different sequences of the 3 items are possible.

To answer this question, we can use the fundamental counting rule or principle, which mathematically says in a sequence of n number of events, where the first event has k1 number of possibilities, and the second event has k2 number of possibilities and so on, the total number of possibilities of the sequence will be, k1 times k2 times k3 times dot dot dot kn.

What this is saying in simpler terms is that when you have a certain number of events, like in our example, shirt, shorts and shoes, to find the total amount of possibilities of the sequence, you multiply the number of ways one event can occur times the number of ways the other events can occur.

So, 7 different colors for the shirt, 4 different colors for the shorts and 5 different colors for the shoes. When we multiply them together, 7 times 4 times 5, we get 140 possibilities of the sequence.

To show this visually, here is a tree diagram for a coin flip and a die roll. You can have a head or a tail on the coin flip, and a 1 through six on the die roll. As you can see on the right, there are 12 possible outcomes.

Using the fundamental counting rule, with the coin flip as event 1 and the die roll as event 2, we have 2 possibilities for the coin flip and 6 for the die roll, so 2 times 6 equals 12, same result as the tree diagram.

One more example is a 4 number pin code for a credit card. Numbers 0 through 9 can be used. So, 4 number code means 4 events and 0 through 9 means 10 possibilities for each event. 10 times 10 times 10 times 10 equals 10,000. So, 10,000 possibilities of the sequence.

Timestamps
0:00 Example Problem Set Up
0:14 Example Data Summarized
0:28 What Is The Fundamental Counting Rule?
0:41 Formula For Fundamental Counting Rule
1:18 Proof Of Fundamental Counting Rule
1:42 Pin Code Example Problem