(upbeat music)
- Hello, I am Professor Rick Jerz,
and I welcome you to this session titled
Discrete Probability Distributions.
This session continues our exploration
into probability distributions
and, in particular, discrete probability distributions,
which are for counted data.
The reason we begin
with discrete probability distributions
is that they are often easier to understand
than continuous probability distributions.
In this session, we will look at one data scenario
that produces a special discrete probability distribution
called the binomial probability distribution.
We will see that the probability concepts
that we learned in our last session
get a little more complex.
We will begin seeing how equations and tables
have been devised to help.
Yep, in our next session, we will learn
how to use Excel, the more modern approach
to solving binomial probability distribution problems.
I think you are going to enjoy this topic.
Let's get started!
Our goals in this session are to define the terms
probability distribution and random variable
to distinguish between discrete and continuous
probability distributions to calculate the mean,
variance, and standard deviation
of a discrete probability distribution
and to describe the characteristics of
and compute probabilities using
the binomial probability distribution.
What is a probability distribution?
A probability distribution is a listing
of all the outcomes of an experiment
and the probability associated with each outcome.
This slide shows an experiment of tossing a coin
three times and observing the number of heads.
Given this data, we are able to ask the question,
what is the probability distribution
for the number of heads?
Although, you might think that this
is a brand new topic we are addressing,
I want to remind you that we have
already seen probability distributions.
Let me show you.
Going back to our students' M&M data,
we counted how many green M&Ms
occurred in each student's bag.
We plotted this and created a frequency distribution.
We converted this data into percentages
by dividing by the total number of M&Ms.
In representing this data as percentages,
we have created a probability distribution.
And in fact, this is a discrete probability distribution
because we could only count whole number of green M&Ms.
We could not find a bag, for example,
with 3 1/2 green M&Ms.
Well, there we have it,
a discrete probability distribution.
Here's another example from our last session on probability,
the probability of what might happen
when we toss a single die.
Remember that once we have these probability distributions
we are able to ask a variety of questions.
For this one, we might wanna know
what is the probability of throwing a three or a four?
That would be the area under the curve,
represented by a three and the four.
The area under the three is 16.7%,
and the area under the four is 16.7%.
So this probability of a three or a four
would be the addition of these two numbers,
or 33.3%, or .33, which is 1/3.
If we return to our probability distribution for green M&Ms,
we could also ask a variety of questions,
like what is the probability of a bag
containing exactly five green M&Ms?
This is the area under the curve represented by five,
which is 16.7%.
Let's say we wanted to create a probability distribution
for tossing two dice.
What we would have to do is tally up
all of the possible outcomes and how often they might occur
and then create the probability distribution
as shown in this chart.
Notice that throwing a seven with two dice
is the most likely outcome, 17%.
If you were betting somebody on the outcome
of throwing two dice, I would bet
that the outcome would be seven
because the probability of a seven
is higher than any other value.
A probability distribution can help you
make good business decisions.
The game of backgammon involves
tossing two dice and moving your pieces around a board.
It turns out that if you understand
the probability distribution of tossing two dice,
you can play a better game of backgammon.
You can win more often.
This is exactly what we wanna do in business.
We wanna make good business decisions
so that we can win more often than lose.
Now that we have reviewed some probability distributions
that we have seen in the past,
we can go back to tossing our coin three times
and create a probability distribution
for the number of heads.
What I'd like you to realize about all
of these probability distributions is that we have to know
something about the experiment,
and we have to be able to count up
all of the ways that the outcomes might occur.
So far for the examples that we have used,
this is not that hard.
Let's look at some characteristics
of all probability distributions.
The probability of a particular outcome
is always between zero and one, or zero and 100%.
The outcomes are mutually exclusive events.
For our toss of a die, you cannot toss both a three
and a four on one die.
It's either a three or a four.
These are mutually exclusive events.
The list is also exhaustive.
The sum of the probabilities of the outcomes
is equal to one, or 100%.
In other words, something has to happen.
Another concept with probability distributions
is that of a random variable,
which is a quantity resulting
from an experiment that by chance
can assume different values.
Let me say this a little bit differently
using our toss of a single die.
We know that the probability of tossing
any single number is equally probable
to tossing any other number.
However, this is not saying that if we throw a die
six times that all six numbers will occur.
Tossing a die is a random event,
and we cannot predict the next value
based upon the last value.
What we can predict that if we toss a die
many, many times, let's say a million times,
and then tally up the results,
that all the numbers will occur approximately equally.
Remember that this is the law of large numbers
that we learned about
in our previous session on probability.
Our examples so far have been for discrete random events,
but there are two types of random variables,
discrete and continuous.
Discrete random variables can only assume
a certain, clearly separated value.
It is usually the result of counting something.
Continuous random variables can assume
an infinite number of values given within a range.
It is usually the result of some type of measurement.
In this session, we concern ourselves
with discrete random events.
In future sessions, we will look
at continuous random events.
Here are some examples of discrete random variables:
the number of students in a class,
the number of children in a family,
the number of cars entering a carwash in an hour,
the number of home mortgages approved
by Coastal Federal Bank last week.
Here are some examples of continuous random variables:
the distance a student travels to class,
the time it takes an executive to drive to work,
the length of an afternoon nap,
the length of time of a particular phone call.
The characteristics of a discrete probability distribution
are nearly identical to what we described earlier
for all probability distributions.
The main features are that the sum
of the probabilities of the various outcomes is one.
The probability of a particular outcome
is between zero and one,
and the outcomes are mutually exclusive.
But for discrete probability distributions,
we can add one more characteristic,
that the item being measured can assume
only certain separated, or counted, values.
When we plot the count of something on the x-axis,
remember that we have at least interval or ratio data,
and, therefore, we can calculate the mean
and standard deviation of these types of distributions.
For the mean of a probability distribution,
we can note two things:
the mean is a typical value used to represent
the central location of a probability distribution,
and also the mean of a probability distribution
is also referred to as its expected value.
This slide shows the equation to calculate the mean,
or expected value, of a discrete probability distribution.
To many, this may appear as a brand new equation,
and it may look a little bit strange.
But let me assure you that we have seen this before.
It is similar to the weighted mean equation
that we saw in an earlier session.
You might recall that we used this weighted mean calculation
when somebody had provided us categorized data
instead of the raw data.
Well, this is exactly what's happening
with discrete probability distribution data.
Somebody has already categorized it
and given us the percent for each category.
If we were to take this weighted mean equation
and recognize that the sum of all weights
is equal to one, the denominator becomes one,
and it goes away.
This would leave us an equation that would essentially be
the addition of the weight of each category
times the value that we are measuring.
This is the mean of a discrete probability distribution.
If you are having a hard time understanding
or visualizing this, don't worry.
The calculation will be done in our Excel model.
We can also calculate the variance and standard deviation
of a discrete probability distribution.
I show the equation for calculating the variance
in this slide.
It, too, may look a little strange or difficult to you,
but this is also similar to what we did
when we calculated the variance of weighted data.
Remember that the standard deviation
is just found by taking the square root of the variance.
The next couple overheads illustrate calculating
the mean variance and standard deviation for an example.
The example is that John Ragsdale sells new cars
for Pelican Ford.
John usually sells the largest number of cars on Saturday.
He has developed the following probability distribution
for the number of cars he expects to sell
on any particular Saturday.
We're given the number of cars
and the probability of selling that quantity.
This slide shows how we would manually
calculate the mean of this probability distribution.
Calculating the variance and standard deviation
by hand gets to be a little bit more difficult.
We aren't going to worry about calculating these by hand.
We will use our Excel model.
Okay, so now we understand some the basics
of probability distributions.
Let's see if we can apply this to another class of problems
that seem to occur frequently in real life.
We are going to learn
about the binomial probability distribution,
which is a discrete probability distribution.
Its characteristics are as follows:
there are only two possible mutually exclusive outcomes
on a particular trial of an experiment,
the random variable is the count of successes
in a fixed number of trials,
the probability of success is the same for each trial,
and each trial is independent of any other trial.
Here are some examples: events that result
in a yes or no, a true or a false, on or off,
correct or incorrect.
The binomial distribution deals with sequential events.
You might recall in our session on probability
that we used tree diagrams to illustrate sequential events.
The limbs of this tree diagram illustrated all
of the outcomes that might occur on any particular event.
For the binomial distribution,
this tree diagram becomes simpler.
Each limb of the tree only has two other limbs
that are possible.
This is because the binomial distribution
can only have two possible outcomes for any experiment.
Well, it turns out that for some very simple experiments
that fit the binomial distribution,
we can tally all the possible outcomes
and count them by hand,
but as our experiment becomes a little bit more extended
or difficult, this hand tallying becomes very difficult.
It turns out that instead of having to tally
everything by hand, mathematicians have derived a formula
that can be used to calculate the probability
of any particular event
for a binomial probability distribution.
I show this equation to you in this slide,
noting the following.
C denotes a combination.
This is the combination from our last session
where we were talking about combinations and permutations.
N is the number of trials.
For example, how many times
are we going to throw a single die?
X is the random variable defined as the number of successes.
If, for example, we throw a single die 36 times,
x might be the number of sixes
that we might observe in these 36 times.
Pi is the probability of success on each trial.
When we toss a single die,
the probability of throwing a six is 16.7%,
so pi, in this case, is .167.
Let's look at a couple examples.
You might not see our M&M data as binomial data,
but I can ask a question that would
make this data be binomial.
For something to be binomial,
we have to have two possible results.
I can configure my question so that there are
only two results.
One condition would be that there are no brown M&Ms.
The other one is that there are brown M&Ms.
There we have it.
There are some brown M&Ms in a bag, or there are not.
Yes or no.
From our collected data, we can also tally the probability
of having a bag with no brown M&Ms
versus the probabilities of having
more than zero M&Ms in a bag.
In our data, the probability of having at least one
or more brown M&Ms in a bag was 86%.
There are many questions that we can ask
about our experiment now that we know this.
What is the probability that two out of four bags
have brown M&Ms?
How about three out of four bags having brown M&Ms?
How about between two and three out of four bags
having brown M&Ms?
In these questions, you can see that the probability,
or pi, remains to be .86.
The number of bags we're concerned about is four,
and we're trying to figure out the probability
of a particular event occurring.
Here's another example from the author.
There are five flights daily from Pittsburgh
via US Airways into Bradford Regional Airport, Pennsylvania.
Suppose the probability that any flight arrives late is 20%.
What is the probability that none of the flights
are late today?
If we were to tally this by hand,
it would take us a little while,
but the author shows that by using the binomial formula,
the calculation is much quicker.
Now, don't worry, we are going to do
these calculations using Microsoft Excel.
Remember, we said that you can always calculate the mean
and variance of a discrete probability distribution.
It turns out that we can do this
for the binomial probability distribution,
and the equations are as follows.
This next slide shows calculating the mean
and variance for the data from our previous example.
The binomial probability formula is complex
for some people to calculate.
If you go back about, oh, let's say 40 years ago
when we did not have calculators,
this binomial formula would be
extremely difficult to calculate.
Therefore, statisticians provided a simpler way
to perform this calculation.
I call this the table approach.
In the back of most probability and statistics textbooks,
you will find a series of binomial probability tables.
I will be giving you a Microsoft Excel model
to perform binomial probability calculations,
and this is what I really prefer.
But I wanna take you to an example
using this table approach, just to give you
a perspective on how it used to be done.
Here's the example.
5% of the worm gears produced by an automatic,
high-speed Carter-Bell milling machine are defective.
What is the probability that out of six gears
selected at random that none will be defective?
Exactly one?
Exactly two?
Exactly three, four, five?
Exactly six out of six?
Since we have asked about probabilities dealing
with six gears, we want to go back
to the probability table where n equals six.
Once we find a table, we want to go
to the column that represents our data.
In this case, we were told that 5%, or .05%
of worm gears are historically defective.
As we look at the values going down this column,
they represent the probability that exactly zero or one
or two or three gears out of the six will be defective.
So in other words, the probability that exactly one gear
is defective out of the six is .232.
For the binomial probability distribution,
we need a series of tables
because the probability distribution
for the binomial function differs depending upon n and pi.
This figure shows a variety
of binomial probability distributions
where we keep the number n constant and vary pi.
This next figure shows just the opposite.
We keep pi constant and vary n.
And you can see that
the probability distributions do change.
A problem that this particular probability distribution
presents to publishers is how many tables
to include at the back of the book.
You might notice that there is not a table
for n equals 42 or 38.
Historically, if you found yourself needing
binomial tables for all values,
you would actually go out to a bookstore
and buy a book, oh, maybe 1,000 pages or so
that had tables going up to n equals 1,000.
The modern calculator eliminated the need
for these big books.
And now Microsoft Excel eliminates
the need for the calculator.
Let me mention another problem associated with these tables.
The tables provide the probability of exactly one value
occurring out of the total number.
Sometimes the question that you might have
involves a less than or greater than question.
In our gear example, we might be curious to know
what is the probability of getting more than four
defective gears out of six.
When we ask these kinds of questions, we must deal
with cumulative binomial probability distributions.
Here is an example that might
help us understand the problem.
A study in June 2003 by the Illinois Department
of Transportation concluded that 76.2%
of front seat occupants used seat belts.
A sample of 12 vehicles is selected.
What is the probability the front seat occupants
in at least seven of the 12 vehicles are wearing seat belts?
In this case, we want the probability of x
being greater than or equal to seven out of 12,
with pi equaling .762.
We can do this, but we have to add up the probabilities
for x equals seven, x equals eight, x equals nine,
x equals 10, x equals 11, and finally x equals 12, wow.
This takes a little bit longer to do.
In our next session, we will see that Microsoft Excel
provides a binomial function.
Calculating cumulative binomial distribution probabilities
will be much easier in Microsoft Excel.
I can't wait!
(upbeat music)
Thanks for taking time to spend some moments with me.
I hope that you have enjoyed what I have shared with you.
If you ever have any questions or need my help,
please let me know.