In essence, Binomial events are a sequence
of identical Bernoulli events.
Before we get into the difference 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 success of 0.6 on each individual trial”.
Additionally, we can express a Bernoulli distribution
as a Binomial distribution with a single trial.
Alright!
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 problems.
In this case, guessing a single true or false
question is a Bernoulli event, but guessing
the entire quiz is a Binomial Event.
Alright!
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 + 1”-many bars - one for each unique
value from 0 to n.
For instance, we could be flipping the same
unfair coin we had from 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 “1 minus p”.
In order to get our favoured 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 favourable.
But these are actually the “combinations”
we already know!
For instance, If we wish to find out the number
of ways in which 4 out of the 6 trials can
be successful, it is the same as picking 4
elements out of a sample space of 6.
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 favourable.
For instance, there are 3 different ways to
get tails exactly twice in 3 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 “1 - 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 3 times during the 5-work-day
week.
If we wish to use the probability distribution
formula, we need to plug in 3 for “y”,
5 for “n” and 0.6 for “p”.
After plugging in we get: “number of different
possible combinations of picking 3 elements
out of 5, times 0.6 to the power of 3, times
0.4 to the power of 2”.
This is equivalent to 10, times 0.216, times
0.16, or 0.3456.
Thus, we have a 34.56% of getting exactly
3 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!
Alright!
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.
After computing the expected value, we can
finally calculate the variables.
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 1 minus p”.
If we plug in the values from our stock market
example, that gives us a variance of 5, 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.
