Binomial Distribution Calculations.

In the last video,

we learned that the binomial distribution is used for modeling

a discrete random variable, x,

that can take on exactly two states,

a set number of trials, n,

a constant probability of "success", p, among the trials.

One application of the binomial distribution,

that is common to six sigma professionals,

is determining the probability

of obtaining a certain number of defective items,

in a sample of a particular size,

when we know the percent of defective items in the population.

In this case,

x, is the number of defective items,

n, is the size of the sample

and p is the percent of defective in the population.

But there is a small problem here,

as you take your sample from the population,

it changes the percentage of defects

in the remaining population.

For example, if there had been five defects

in a population of 100,

which is five percent,

then if the first item you take out to sample is a defective item,

it leaves four defective items in a population of ninety nine,

which is closer to four percent.

So, to ensure that the probability of success

remains constant enough in the sample,

we use the following rule of thumb.

The sample size, n,

has to be less than ten percent of the population size

and the population size must be at least 50.

This ensures that the third requirement

of the constant probability of success is satisfied.

If those assumptions are true,

then the probability of x defective items

can be modeled with the binomial formula.

The probability of x, is equal to n factorial,

divided by x factorial, times and minus x factorial,

p raised to the x, one minus p raised to the n minus x.

X is the number of successes in the experiment,

n is the number of trials and p is the probability of success.

The exclamation mark is pronounced factorial,

and most calculators have a factorial button.

Let's do an example.

Suppose that a sample of size four,

is randomly chosen from a batch of size 100,

that is known to be five percent effective.

What is the probability,

that there is exactly one defective item in your sample?

In this example,

we are looking at the probability of exactly one defect.

So x equals one.

The sample size is four

and the probability is 0.05,

for the five percent defective.

Note that there are two states for x,

defective versus non-defective.

There are a set number of trials, four,

and the population is larger than 50,

and the sample size of size four

is less than ten percent of the population.

So all of our assumptions are matched

to use the binomial formula.

We then plug x, n and p into the formula

and obtain an answer of 0.1715.

This problem can be solved very easily in excel using the following formula.

P of x equals binom dot dist, x, n, p

and then a cumulative indicator.

Into a cell in excel,

you would type exactly what is in the parentheses,

equals binom dot dist, one comma four,

comma point zero five, comma false.

You can replace false with true,

if you want the probability

that x is less than or equal to the chosen x,

rather than exactly equal to x.