11 1binomial

# 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 Probability Of Success: pos = 1/6 … (in fraction or decimal form)

… … Click OK

… …(You should get 0.0154)

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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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