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Fundamental Counting Principle

Before learnign about probability, it helps to learn about what is known as the ‘fundamental counting principle, and observe some fundamental counting principle examples.

It is something that is useful when you want to work out the number of possible combinations from a selection of different choices/outcomes.

Fundamental Counting Principle:

If there were &nbsp “a” &nbsp possible ways of performing one action,

and &nbsp “b” &nbsp possible ways of performing a second action,

then there would be &nbsp a × b &nbsp ways of performing &nbsp’both’&nbsp actions together.

This principle can often be especially helpful when dealing with probability, as we area able establish how many outcomes may be possible with certain events.

Provided that the outcome or options are “independent” of each other, and choosing one option wouldn’t have an affect the other.

Fundamental Counting Principle Examples


Image of clothes selection to show the Fundamental Counting Principle

Let’s say a person is getting dressed in the morning.

For the shirt they can wear a &nbsp”RED“&nbsp shirt or a &nbsp”GREEN“&nbsp shirt.
( 2&nbsp possible choices )

For the trousers they can wear &nbsp”BLACK“&nbsp or &nbsp”WHITE“&nbsp or &nbsp”BLUE“&nbsp trousers.
( 3&nbsp possible choices )

The first action &nbspa,&nbsp is to choose the shirt colour.

The second action &nbspb,&nbsp is to choose the trouser colour.

a × b &nbsp = &nbsp 2 × 3 &nbsp = &nbsp 6

There are &nbsp6&nbsp combinations that can be made in the ways to choose the colour of the shirt and trousers.

Red Shirt with White Trousers , Red Shirt with Black Trousers , Red Shirt with Blue Trousers

Green Shirt with White Trousers , Green Shirt with Black Trousers , Green Shirt with Blue Trousers


The fundamental counting principle can also be used when there are more than &nbsp2&nbsp different events happening.

A boy wants to buy &nbsp3&nbsp video games at a shop.

They want to buy:

a &nbsp “SOCCER” &nbsp game,
a &nbsp “SHOOTING” &nbsp game,
and a &nbsp “SIMULATION” &nbsp game.

At the shop there is a selection of:

4&nbsp Soccer games &nbsp , &nbsp 12&nbsp Shooter games &nbsp , &nbsp 8&nbsp Simulation games.

Of these options, how many combinations of games can be selected?

This case with video game is similar to example &nbsp(1.1),&nbsp but this time there are &nbsp3&nbsp events happening,&nbsp 3&nbsp different video game options being seleceted together.

So instead of &nbsp a &times b, &nbsp we will have &nbsp a × b × c.

4&nbsp ways to select a Soccer game &nbsp ( a )

12&nbsp ways to select a Shooting game &nbsp ( b )

8&nbsp ways to select a Simulation game &nbsp ( c )

a × b × c &nbsp = &nbsp 4 × 12 × 8 &nbsp = &nbsp 384

There are &nbsp9600&nbsp combinations of the &nbsp3&nbsp video game selections specified in the shop, when the boy buys a separate “Soccer”, “Shooting” and “Simulation” game together.

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