# The Binomial Distribution (or Bernoulli Distribution)

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The **Binomial Distribution** is a common example of a **discrete probability distribution**.

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The **Binomial Distribution** is often called the **Bernoulli Distribution**.

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## Binomial Sequences (or Bernoulli Sequences)

- Each experiment has only
**two**outcomes (**success**or**failure**)

- Each experiment is called a
**Trial**or a Bernoulli Trial .

- Each outcome is
**independent**of previous trials.

- The probability of each outcome does not change between trials.

In a Bernoulli Sequence, the number of successes follows the **binomial distribution**.

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## Notation

**X**represents a random variable that has a binomial distribution.

**n**= number of trials in the sequence

**p**= probability of success in each trial

**q**= probability of fail in each trial $\big( q = 1 - p \big)$

**X ~ Bi(n, p)**… or …**X ~ B(n, p)**

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### Example 1a

A fair 6 sided die is rolled **4** times.

Let **X** represent the number of times a **3** is rolled.

… … ie a success is **3**, a fail is anything other than **3**

Express the variable **X** in Standard Notation.

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**Solution**

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… … **Note:** **X** has a binomial distribution because

… … … **1** .. each trial is **independent** and

… … … **2** .. there are only **2** possible outcomes (success or fail).

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… … $n = 4$ … … there are **4** trials (we roll the die **4** times)

… … $p = \dfrac{1}{6}$ … … the probability of success $Pr(X = 3) = \dfrac{1}{6}$

… … so

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… … $X \sim Bi \Big( 4,\; \dfrac{1}{6} \Big)$

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## The Binomial Distribution

Given that **X ~ Bi(n, p)**, the binomial distribution states that:

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### Example 1b

A fair die is rolled **4** times.

**X** represents the number of times a **3** is rolled.

ie: $n = 4, \; p = \dfrac{1}{6} \qquad so \;\; X \sim Bi \Big(4,\; \dfrac{1}{6}\Big)$

Create a table showing the probability distribution for **X** correct to 4 decimal places

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**Solution**

… … The sample space for **X** is: $X = \big\{0,\; 1,\; 2,\; 3,\; 4 \big\}$

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… … The probability for each possible outcome can be calculated using the Binomial Distribution

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… … For example, the probability that we get **3** successes out of **4** rolls is

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… … $Pr(X = 3)$ … so $n = 4, \; p = \dfrac{1}{6}, \; q = \dfrac{5}{6}, \; x = 3$

… … $Pr(X = 3) = ^4C_3 \Big( \dfrac{1}{6} \Big)^3 \Big( \dfrac{5}{6} \Big)^1$

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… … … … $= \Big( 4 \Big) \Big( \dfrac{1}{216} \Big) \Big( \dfrac{5}{6} \Big)$

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… … … … $= \dfrac{20}{1296}$

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… … … … $\approx 0.0154$

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… … The probabilities for the other outcomes can be calculated in the same way.

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… … The entire distribution for **X** is:

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## Binomial Distribution on the CAS calculator

From the **Main** screen.

Go to the **Interactive** menu and select **DISTRIBUTION** (near the bottom)

From the second menu, select **BINOMIAL PDF**

If we want to find $Pr(X = 3)$ given $X \sim Bi \Big( 4,\; \dfrac{1}{6} \Big)$

… … Enter **x = 3**

… … Enter **trials = 4**

… … Enter **P**robability **O**f **S**uccess: **pos = 1/6** … (in fraction or decimal form)

… … Click **OK**

… …(You should get **0.0154**)

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### CAUTION: Do not use calculator notation in your written answers.

If you need to show working, stick to correct mathematical notation.

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## When **NOT** to use the binomial distribution

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If a specific **order** is required do **NOT** use the binomial distribution.

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- The outcomes are
**independent**so the probability of a specific result is the**product**of the required outcomes.

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### Example 3

Given the situation above, find the probability that the first roll is a success (a 3) followed by 3 fails (not 3) correct to 4 decimal places.

**Solution**

… … The order required is given so do **not** use binomial distribution

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… … $Pr(SFFF) = \dfrac{1}{6} \times \dfrac{5}{6} \times \dfrac{5}{6} \times \dfrac{5}{6}$

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… … … … $= \dfrac{125}{1296}$

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… … … … $\approx 0.0965$

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## Graphs of Binomial Distributions

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### Positively Skewed (p < 0.5)

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Consider the earlier distribution: $X \sim Bi \Big( 4,\; \dfrac{1}{6} \Big)$

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This graph is **positively skewed** (the long tail is at the **positive** end).

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A binomial distribution graph will be **positively skewed** if $p < 0.5$

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- This is a probability distribution so the sum of the probabilities must be 1.
- This means the sum of the columns in the graph will be 1.

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- Because this is a
**discrete**distribution, we draw the graph as a histogram- Remember that a histogram has gaps between the columns

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### Normal Distribution (p = 0.5)

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Large values of **n** gives a shape approximating a continuous smooth curve with a distinctive **bell** shape.

Source: Wikipedia (Normal Distribution)

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Compare the shape of the graph to the shape of a traditional church bell.

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** Note** the **sum** of the columns in a binomial distribution graph is still **1**.

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- The
**area**under the binomial distribution curve is equal to 1.

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We will encounter this bell shaped curve in **Chapter 12: Normal Distribution** which involves **continuous** variables.

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