11Abinomialdist


 * The Binomial Distribution ** ** (or Bernoulli Distribution) **

This is a common example of a discrete probability distribution.

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

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.

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 ( q = 1 – p )

X ~ Bi(n,p) ... or ... X ~ B(n, p)


 * Example 1a **

A fair die is rolled 4 times.
 * A fair die is a dice with numbers {1, 2, 3, 4, 5, 6} that is not biased so each number has an equal probability of appearing

Let X represent the number of times a 3 is rolled.
 * (ie a success is 3, a fail is anything other than 3)

Note that X has a binomial distribution because
 * each trial is independent and
 * there are only 2 possible outcomes (success or fail).

math \\ . \qquad n = 4 \\. \\ . \qquad p = \dfrac{1}{6} \qquad. math

so

... ... X ~ Bi(4, 1/6)

The Binomial Distribution

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


 * Example 1b **

A fair die is rolled 4 times. X represents the number of times a 3 is rolled. ie: n = 4, p = 1/6 so X ~ Bi(4, 1/6)

The probability that we get 3 successes out of 4 rolls is math \\ . \qquad Pr(X=3) = ^4C_3 \Big( \dfrac{1}{6} \Big)^3 \Big( \dfrac{5}{6} \Big)^1 \\. \\ . \qquad \qquad \quad \quad = \Big( 4 \Big) \Big( \dfrac{1}{216} \Big) \Big( \dfrac{5}{6} \Big) \qquad. \\ . \\ . \qquad \qquad \quad \quad = \dfrac{20}{1296} \\. \\ . \qquad \qquad \quad \quad = \dfrac{5}{324} \\. \\ . \qquad \qquad \quad \quad \approx 0.0154 math

The entire distribution for this Variable is.

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 ~ Bi(4, 1/6) Enter x = 3 Enter trials = 4 Enter prob = 1/6 (in fraction or decimal form) (You should get 0.0154)

{In some older versions of the Classpad, go to the **Statistics** package, then select Distributions from the **CALC** menu}

When NOT to use the binomial distribution

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

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


 * Example 3 **.

Given the situation above, the probability that the first roll is a success (a 3) followed by 3 fails (not 3) is given by:

math \\ . \qquad Pr(SFFF) = \dfrac{1}{6} \times \dfrac{5}{6} \times \dfrac{5}{6} \times \dfrac{5}{6} \qquad. \\ . \\ . \qquad \qquad \qquad \quad = \dfrac{125}{1296} \\. \\ . \qquad \qquad \qquad \quad =0.0965 math

Graphs of Binomial Distributions

Consider the earlier distribution: X ~ Bi(4, 1/6)

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

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

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.

Because this is a discrete distribution, we draw the graph as a __**histogram**__ (with __gaps__ between the columns)



Normal Distribution (p = 0.5)

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

Compare the shape of the graph to the shape of a traditional church bell. Notice that the sum of the columns in each graph is still 1.

We can, therefore, say that for the bell shaped curve, the area under the curve is equal to 1.

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