4. Probability and the Binomial Distributions

# 4.1 Probability

The basic definition of probability is a ratio of things you can count (a ratio of their frequencies) :

(4.1)

where

is the probability that event happens,

is the number of ways can happen and

is the total number of outcomes (all possibilities).

**Example 4.1** : What is the probability of drawing a queen from a deck of cards :

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To use mathematically we set

Where, probability-wise:

0 means definitely will not occur, and

1 means definitely will occur.

This is a method we can use instead of using percent. To compute probabilities, we first need to know how to count.

**Fundamental Counting Rule **

Say you have n events in order, and for event there are ways for it to happen. Then the number of ways for the events to play out is :

(The giant pi symbolizes a multiplication convention in the same way that a giant sigma symbolizes a summation convention as described in Section 1.3.)

**Example 4.2** How many combinations are there on a lock with 3 numbers?

Lay out the events as : , , and . Note that each number can be anything from 0 to 9 giving 10 possibilities () for each event. So the number of possible lock combinations is

Note that you could have guessed this because the combination range from 000 to 999 — counting in base 10.

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**Example 4.3** Suppose that a hardware store can produce paints with the following qualities :

Colour : red, blue, white, black, green, brown, yellow (7 colours)

Type : latex, oil (2 types)

Texture : flat, semigloss, high-gloss (3 textures)

Use : indoor, outdoor (2 uses)

How many ways are there to combine these qualities to produce a can of paint?

* Answer* : From the above list and the number of possible paint kinds is:

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**Applications of the Fundamental Counting Rule**

We are interested in applying the fundamental counting rule to two special, important cases :

- Permutations.
- Combinations.

Let’s define each one.

**Permutations**.

The number of ways, or permutations, of selecting objects from a collection or objects, *while keeping track of the order of selection* is ^{[1]}

This formula follows from the fundamental counting rule. With objects there are ways to select the first object. After selecting the first object there are ways to choose the second object so , etc. up to :

**Example 4.4** : How many ways are there to choose 5 numbered balls from a bucket of 25 to make a lottery number?

* Answer* : possibilities.

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** 2. Combinations**.

The number of ways of selecting objects from a collection of objects *without* caring about the order is :

That last symbol is colloquially called “ choose ”. The second last expression demonstrates the application of the fundamental counting principal, it says

where is just the number of ways of arranging objects while caring about the order, .

As a practical matter, never try to compute It will usually be unimaginably big. Use the formula that directly shows the fundamental counting rule as shown in the following example.

**Example 4.5** : How many ways are there to select 10 balls from a bucket of 100?

* Answer* :

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The symbol is also known as the *binomial coefficient* because it shows up in algebra when you expand expressions of the form . For example^{[2]}

The binomial coefficients can be quickly computed using Pascal’s triangle :

Referring to Pascal’s triangle we can quickly write

for example.