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Instructor: Welcome back.

In the last video we mentioned binomial distributions.

In essence, binomial events are a sequence

of identical Bernoulli events.

Before we get into the differences

and similarities between these two distributions

let us examine the proper notation

for a binomial distribution.

We use the letter B to express a binomial distribution

followed by the number of trials

and the probability of success in each one.

Therefore, we read the following statement as

variable X follows a binomial distribution with 10 trials

and a likelihood of 0.6 to succeed on each individual trial.

Additionally, we can express a Bernoulli distribution

as a binomial distribution with a single trial.

All right, to better understand the differences

between the two types of events

suppose the following scenario.

You go to class and your professor

gives the class a surprise pop quiz

which you have not prepared for.

Luckily for you, the quiz consists

of 10 true or false questions.

In this case, guessing a single true

or false question is a Bernoulli event,

but guessing the entire quiz is a binomial event.

All right, let's go back

to the quiz example we just mentioned.

In it, the expected value of the Bernoulli distribution

suggests which outcome we expect for a single trial.

Now, the expected value

of the binomial distribution would suggest the number

of times we expect to get a specific outcome.

Great. Now, the graph

of the binomial distribution represents the likelihood

of attaining our desired outcome a specific number of times.

If we run n trials, our graph

would consist n plus one many bars

one for each unique value from zero to n.

For instance, we could be flipping the same unfair coin

we had from the last lecture.

If we toss it twice, we need bars for the three

different outcomes; zero, one, or two tails.

Fantastic. If we wish to find the associated likelihood

of getting a given outcome a precise number of times

over the course of n trials, we need to introduce

the probability function of the binomial distribution.

For starters, each individual trial is a Bernoulli trial

so we express the probability of getting our desired outcome

as p and the likelihood of the other one as one minus p

in order to get our favorite outcome exactly y many times

over the n trials, we also need to get the alternative

outcome n minus y many times.

If we don't account for this, we would

be estimating the likelihood of getting our desired outcome

at least y many times.

Additionally, more than one way to

reach our desired outcome could exist.

To account for this, we need to find the number

of scenarios in which y out of the n

many outcomes would be favorable.

But these are actually the combinations we already know.

For instance, if we wish to find

out the number of ways in which four

out of the six trials could be successful,

it is the same as picking four elements

out of a sample space of six.

Now you see why combinatorics are a fundamental part

of probability; thus, we need

to find the number of combinations

in which y out of the n outcomes would be favorable.

For instance, there are three different ways to

get tails exactly twice in three coin flips.

Therefore, the probability function

for a binomial distribution is the product

of the number of combinations of picking

y many elements out of n times

p to the power of y

times one minus p

to the power of n minus y.

Great. To see this in action, let us look at an example.

Imagine you bought a single stock of General Motors.

Historically, you know there is a 60% chance the price

of your stock will go up on any given day

and a 40% chance it will drop.

By the price going up, we mean that

the closing price is higher than the opening price.

With the probability distribution function

you can calculate the likelihood of the stock price

increasing three times during the five workday week.

If we wish to use the probability distribution formula

we need to plug in three for y, five for n and 0.6 for p.

After plugging in, we get number of different possible

combinations of picking three elements out of five times 0.6

to the power of three times 0.4 to the power of two.

This is equivalent to 10 times 0.216 times 0.16

or 0.3456

Thus, we have a 34.56% chance of getting exactly

three increases over the course of a work week.

The big advantage of recognizing the distribution is

that you can simply use these formulas and plug

in the information you already have.

All right, now that we know the probability function

we can move on to the expected value.

By definition, the expected value equals the sum

of all values in the sample space multiplied

by their respective probabilities.

The expected value formula for a binomial event

equals the probability

of success for a given value

multiplied by the number of trials we carry out.

This seems familiar

because this is the exact formula we used when

computing the expected values

for categorical variables in the beginning of the course.

After computing the expected value

we can finally calculate the variance.

We do so by applying the short formula

we learned earlier;

Variance of y equals the expected value

of y squared minus the expected value of y squared.

After some simplifications, this results in

n times p times one minus p.

If we plug in the values from our stock market example

that gives us a variance

of five times 0.6 times 0.4, or 1.2.

This would give us a standard deviation

of approximately 1.1.

Knowing the expected value

and the standard deviation allows us to make more

accurate future forecasts.

Fantastic. In the next video,

we are going to discuss Poisson distributions.

Thanks for watching.