The binomial distribution is a discrete distribution displaying data that has only** TWO OUTCOMES** and each trial **includes** **replacement**. The binomial distribution approaches a normal distribution as the sample size increase and therefore this approximation is better for larger sample sizes.

Such as:

- PASS / FAIL
- GO / NO-GO
- SUCCESS / FAILURE
- IN / OUT
- YES / NO
- HOT / COLD
- MALE / FEMALE
- WORKS / DOESN'T WORK
- DEFECTIVE / NOT DEFECTIVE
- RIGHT-HANDED / LEFT-HANDED
- HIGH / LOW
- CORRECT / INCORRECT
- HEADS / TAILS

**Assumptions**

- Each trial has only two outcomes
- The experiment has
*n*identical trials - Each trial is independent of the other trials
- The probability of getting one outcome (success) p is held constant and the probability of getting the other outcome (failure) is also held constant, represented by (1 - p).
- Includes replacement for each trial. It can be used to approximate without replacement trials but it is suggested to use hypergeometric distribution formulas.

Let's evaluate the__assumptions:__A coin flip is an example of two outcomes and each flip has the same chance for both outcomes (meaning independent). Flipping the coin*n*amount of times means there are*n*identical and independent trials. Nothing changes from one flip to another and the odds are the same for each flip.

The following formula is used to compute the number of **experimental outcomes** resulting from x successes in* n* trials.

For example: 4! (4 factorial) = 4*3*2*1 = 24

The **Binomial Distribution Probability Function** is shown below:

A manufacturing process creates 3.4% defective parts.
A sample of 10 parts from the production process is selected. What is
the probability that the sample contains exactly 3 defective parts?__SOLUTION:__

There are two outcomes: Defective / Not-Defective, therefore the Binomial Distribution equation is applied.

p = 0.034

n = 10

then q = 1-p = 1 - 0.034 = 0.966 (96.6%)

Need the probability that x = 3.

Substitute the values into the Binomial Probability Function and solve:

Forty-five percent of all registered voters in a
national election are female. A random sample of 8 voters is selected.
The probability that the sample contains 2 males is:__SOLUTION:__

55% are male and there are two outcomes: MALE / FEMALE

n = 8

p = 0.55 (55%)

q = 1-p = 0.45

Need the probability that x = 2

79% percent of the students of a large class passed
the final exam. A random sample of 4 students are selected to be
analyzed by the school. What is the probability that the sample contains
** fewer** than 2 students that passed the exam?

There are two outcomes: PASS / FAIL

p = 0.79

n = 4

Solving for the probability that x < 2.

Note that we need to compute p(x<2). So, it is necessary to add the P(x=0) to the P(x=1).

Each P(x=0) and P(x=1) must be calculated separately and added.

p(x<2) = p(x=0) + p(x=1)

Substitute the following values into the binomial equation to find p(x=0):

n = 4

p = 0.79

q = 1 - p = 1 - 0.79 = 0.21**x = 0**

PLUS

n = 4

p = 0.79

q = 1 - p = 1 - 0.79 = 0.21**x = 1**

There is 3.12% chance of selecting fewer than 2 students that passed the
exam when randomly selecting 4 students when 79% of them passed.

There are also Binomial tables that can be used when the input variables are known (and found within the table).

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