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.