(upbeat music)
Hello.
I am Professor Rick Jerz,
and I welcome you to this session titled, "Probability."
This topic is always one of the most difficult for students
so I warn you that you need to pay close attention,
and practice solving problems.
The word probability is so scary
that even textbook publishers
avoid it in their textbook titles,
fearing that by including it,
the textbook will not sell well.
As you explore this topic in our textbook,
it might seem to you that there are two topics,
probability concepts, and counting methods.
And they might appear somewhat disjointed.
Our textbook author struggles making this connection.
And so do many others.
I will do my best to connect them.
Let's get started.
We have a number of goals for this session.
We want to understand
some probability concepts and terminology.
These include the terms probability, experiment, event,
and outcome, conditional, and joint probability.
We want to learn how to produce
probability distribution graphs and multiple event graphs.
We want to learn how to calculate probabilities
using rules of addition and rules of multiplication,
we want to recognize probabilities using classical,
empirical, or subjective approaches,
and finally, we want to be able to calculate
the number of arrangements, permutations, and combinations.
I am immediately providing you this graphic
as an overview of this topic,
which will allow us to keep track of what we are learning.
Yes, it may look a little complex,
but I believe that it will be useful.
Businesses make decisions which can vary as shown.
In business, we want to reduce the risk
of making bad decisions.
If we knew exactly what might happen,
business decisions would be easy.
Consider that every business decision
has an economic payoff,
and we want to win more often than we lose.
Actually, we want to win a lot.
Decisions involve something happening.
In statistics, we prefer to use the word experiment
instead of the word something,
maybe because it seems more scientific,
and that it seems to portray
that we might not know the results of a situation,
and experiment until the passage of time.
Experiments can be complex,
such as the sale of a product,
the production of good quality products,
resolving a customer support call,
or they can be simple, such as the rolling dice,
or the tossing of a coin.
Hmm, I like these simply experiments,
they're easy to understand,
so maybe they will be helpful to explore this topic.
In experiments, it is typical
that more than one thing can happen, however, again,
let's pick a better word that multiple things.
Statisticians like the word outcomes.
An experiment can have many outcomes.
For example, when we roll a die,
there can be six outcomes,
a one, a two, three, four, five, or six.
Tossing a coin can have two outcomes, a heads or a tail.
The sale of a product might have many outcomes
from selling nothing to, let's say, a million.
Yep, sales is a complex experiment,
whereas tossing a coin is a simple experiment.
Now, what do you want to know?
We want to know the likelihood of, well,
it can be just about anything.
In our simple tossing of a dice,
we might want to know the likelihood or tossing a six,
or an even number.
In our more complex sales example,
we might want to know the likelihood of selling
between 1,000 and 2,000 products.
Instead of saying we want to know this,
the statisticians use the word event or events.
Yep, more scientific.
In tossing a die, the event of interest
can be tossing a two or a three, or an odd number,
or a number less than four.
As you can see, there can be many events of interest.
So here we have it, an experiment of interest,
all the possible outcomes,
and an event being the collection
of one or more outcomes of interest.
Here is an illustration of these words
being used for the tossing of our dice,
and the selection of a board member.
Now, here's a very powerful concept,
the likelihood of the events of interest,
what should we call this?
Why not likelihood?
Nope, the statisticians call this probability.
By definition, the probability can only assume a value
between zero and one, or zero and 100%.
A value near zero means the event is not likely to happen,
a value near one means it is very likely,
an exact zero means that the even can never happen,
and the value of one means that the event
is certain to happen.
This pie chart showing the number of colored M&Ms in our bag
portrays the idea of probability
in our experiment of picking an M&M out of a bag.
100% is the complete pie,
the color outcomes are wedges of the pie.
Red is the biggest pie wedge and is more likely to occur
than a green color, if I were to bet you, let's say, $100,
about the color of the next M&M, I would bet red.
The pick involves uncertainty, or some randomness,
but red is more likely.
This illustration shows more examples of experiments
and possible probabilities.
Incidentally, this chart was created a few years ago
when the Dow was much lower,
so you can see that even knowing something
about probabilities doesn't mean that we are always correct.
The Dow has already been higher than 26,000.
Some randomness can still affect events' probabilities.
You might notice at this topic of probability
is not completely new to us.
We were introduced to it when we learned
how to graph relative frequencies
by producing pie charts and relative frequency diagrams.
Statisticians prefer bar charts
as a more universal way of showing probabilities.
They then call these diagrams probability distributions,
which becomes another new word or concept to us.
Even though we now have bars instead of wedges of a pie,
it is a probability distribution.
If we add up all the probabilities for each bar,
meaning all the outcomes, we end up with 100%.
This means that something will happen.
Statisticians refer to this as collectively exhaustive.
Another thing portrayed is that each outcome is unique.
We cannot have an M&M be both
red and yellow at the same time.
Statisticians say that the outcomes are mutually exclusive,
an event of interest can still be
a single color or multiple colors,
and to figure out the probability of an event,
we can add up each outcome's area under the bar.
For continuous data, we do not have individual bars,
so I like to describe this approach
to calculating probabilities more generically
as calculating the area under the curve
representing the event of interest.
For example, the probability of picking a red is 26%.
For the probability of picking a red or an orange,
we would add 26% to 16% and come up with 42%.
The probability of picking a primary color is 63%,
this is the addition of blue, yellow, and red bar areas.
How about this interesting event?
The probability of a red and an orange,
this is a tricky question,
since an M&M cannot be both red and orange at the same time,
so this event has a probability of zero.
Yes, a probability can be zero,
be careful of the use of the conjunctions and and or.
Probability distributions are very useful
in visualizing experiments and outcomes.
Out M&M color probability distribution was easy to create,
but creating probability distributions can be harder.
For example, how about creating a probability distribution
for the tossing of two dice?
We could start by tallying all the possible outcomes
and then grouping them as to their result.
For example, a five can occur if the first die is a three
and the second is a two, or reverse,
or a four and a one, or a one and a four.
Hmm, four ways, by tallying,
we can see that there are 36 outcomes
for throwing two dice.
Once we have this probability distribution,
we can now use it to answer a variety of event questions.
What is the probability of throwing a seven or eight?
Well, the area under the curve is 16.7% plus 13.9%
for a total of 30.6%, less than half the time.
Throughout this course, we will use and explore
probability distribution diagrams
to help us calculate probabilities.
Let's get back to our colored M&Ms and ask another question.
What is the probability of picking a color that is not red?
Well, again, we can calculate the probability of this event
by adding the areas of all bars that are not red,
meaning 16 plus 16 plus five plus 16 plus 21, or 74%.
This, however,
takes a little bit of work for our simple question.
Statisticians have come up with some probability rules
that can help us not only understand probabilities,
but often make it faster
and more convenient to calculate probabilities.
The rules that we will explore are the complement,
the rule of addition, and the rule of multiplication.
The complement rule helps us with these not questions.
You can subtract the probability
of the event occurring from one.
Here are two ways to write the rule mathematically
by transposing variables, so in our example,
the probability of not red is 100% minus 26%, or 74%.
Yep, much quicker calculation.
The rule of addition is nothing new,
we have been using it all along.
It simply says that to calculate the probability
for an event that contains more than one outcomes,
you add up each outcome's individual probability.
The probability of a seven or eight in our two dice example
is the probability of a seven
plus the probability of an eight.
Graphically, you can see this as the area
under the curve containing seven and eight.
There is a slight problem, however.
Sometimes people will report probabilities of outcomes
that are overlapping,
meaning that they are not mutually exclusive.
Whenever this happens, you must subtract the overlap.
This general rule of addition includes this subtraction.
Yes, a bit more complex.
Here is an example of this general rule of addition
applied to playing cards.
I should mention that playing cards is much more complex
than tossing a coin or dice.
What is the probability that a card chosen at random
from a standard deck of cards
will be either a king or a heart?
If you are familiar with cards,
you know that a deck contains 52 cards,
with four suits and two colors.
There are four kings, and there are 13 hearts.
Therefore, the probability of drawing a king
is four out of 52, and a heart is 13 out of 52.
However, if you simply add up these two numbers,
you would be wrong.
This is because there is a king of hearts
that has been counted twice, once with the kings,
and again with the hearts.
So the general rule of addition
says to subtract this overlap.
The result is 16 over 52, or .3077, or 30.77%.
You will have a chance to practice this rule
with some self assessment problems.
It can be a little tricky.
Sometimes people like to use Venn diagrams
to illustrate some of these probability rules.
This Venn diagram illustrates joint probability.
The rules of multiplication are a little more difficult.
This rule is used for independent events,
meaning that one event is not connected to the other.
For example, not at the same time.
Again, we have a special rule and a general rule.
When we have two or more independent events
and want to know the resulting probability for all,
we multiply the event probabilities together.
For example, if we toss a dice two times
and ask what is the probability
of throwing a four two times in a row,
we multiply one sixth times one sixth and end up with .028.
If we ask about throwing a four five times in a row,
we would multiply one sixth by itself five times.
The general rule of multiplication takes into account
something that, however, might occur.
Sometimes the second event might be influenced
or conditioned by the outcome of the first.
This is referred to as conditional probability.
When this happens, we must use the conditioned probability
of the second event.
Here is an example.
A golfer has 12 golf shirts in his closet.
Supposed nine of these shirts are white and the others blue.
He gets dressed in the dark,
so he just grabs a shirt and puts it on.
He plays golf two days in a row
and does not do laundry.
What is the likelihood that both shirts selected are white?
On his first random pick, the probability of picking white
is nine divided by 12, pretty simple.
However, the second try is more complex.
By not doing laundry means that the white shirt
is not returned to the group,
so there are only 11 remaining, and of these 11,
had he picked the white on the first try
and not replaced it, there are only eight white shirts left.
Think about this a little.
The second pick has been conditioned by our first pick.
Wow, we have conditional probability,
the probability of picking a white shirt on the second pick
is eight divided by 11.
To answer the original question,
the rule of multiplication is used
for these two sequential events,
we multiply nine over 12 by eight over 11,
and end up with .55.
Yep, this rule of multiplication seems harder,
don't worry, you will have a chance to practice this rule
in our problem self assessments.
This rule of multiplication can apply
even with us picking colored M&Ms from a bag,
we might ask what is the probability of picking a red
and then another red.
An important thing to realize is that
if we return our first pick to the bag,
or return a shirt in our previous example,
we don't have conditional probability.
In our shirt example, we would then multiply nine over 12
by nine over 12.
I want to mention that sometimes people like to diagram
these multiple sequential events with tree diagrams.
Hey, this is great, we get to see our tree diagrams
being applied again, another opportunity to use tree plan.
Let's take a break from all these probability rules
and go back to discuss
some general concepts about probability.
Although it may seem like there are all kinds of ways
of assessing probability, there are really only three ways,
they are as follows:
subjective, empirical, and classical.
Subjective is a guess based upon intuition or judgment,
empirical is based upon sampled data,
classical is based upon a calculation of counting
all the outcomes that satisfy events of interest
divided by a count of all possible outcomes.
Both the classical and empirical approaches are objective,
meaning that data is used,
whereas the subjective approach is not based upon data.
This is not to say that people and managers
make subjective decisions without some data,
it is just that there is not enough data
to analyze statistically.
In business and the focus of this course
is to learn objective techniques.
There are many times where decisions
must be made without data.
This slide and our textbook provides examples
where subjective probability is used.
Later in this course, we will learn about sampling
and how to assess risk based upon samples,
which is empirical probability.
Yes, we can use a sample as a way to understand
what might happen in the big picture.
For example, based upon red M&Ms in our own bag of M&Ms,
what might I say about the production process for all M&Ms?
We created relative frequency diagrams for our M&Ms,
the question becomes how do you use this information
to aid decisions?
When we use samples, the bigger the sample, the better.
This is referred to as the law of large numbers.
Classical probability is an exact and accurate calculation.
It is calculated as the number of outcomes of interest
divided by the total number of possible outcomes.
We will soon see that one challenge
is being able to count all the outcomes.
This graphic provides a summary of these probability types.
Sometimes it might not be clear
as to whether we are using empirical
or classical probability, for example, our colored M&M.
If, for example, we were trying to determine
the likelihood of picking a red M&M
out of all classmates' M&M bags collectively,
this is a classical probability problem
since we have concerned ourselves
with only this class' M&Ms.
If we are trying to use our class M&Ms to predict,
oh, let's say the color of M&Ms
at all Hy-Vee grocery stores,
this would be an empirical probability problem
since we don't have access
to all M&Ms at all Hy-Vee grocery stores.
When people ask probability questions,
it is always important to clarify what the person is seeking
so that you know how to approach the problem.
For example, with the deck of cards,
there can be all kinds of questions.
What is the probability of drawing a king?
A heart?
A king or a heart?
A king and a heart?
A heart if the king is not replaced?
A king followed by a heart without replacement?
A king then a heart if the king is put back into the deck?
Or not a king?
Hoo, this is starting to make my head spin a little.
Let's return to classical probability,
which is the most accurate way to calculate probabilities.
Recall that we divide the number of outcomes of interest,
meaning outcomes for our event,
by the total number of outcomes possible.
Looks pretty simple, however,
this counting can become complex very quickly,
which is why we learn about arrangements, combinations,
and permutations in the last part of this session.
Arrangements, combinations, and permutations
help us count outcomes,
and they are extremely useful methods
if we want to use classical probability calculations.
Sure, sometimes counting is easy for some experiments.
Consider an experiment of rolling a six-sided die.
What is the probability of the event
an even number of spots appear face up?
We know that a die has six sides with three even numbers
and three odd numbers,
so the probability of throwing an even number
is simply three divided by six.
But be careful, this does not say that an even number
will always occur exactly 50% of the time,
randomness still exists, however, after a million tosses,
we would end up with about half of the results
being an even number.
Throwing dice is an easy experiment.
Let's try this one, how many ways can you pick five cards
from a deck of 52 cards?
Yes, tell me very quickly, I'm waiting.
This is much more complex.
We could sit down and try to tally these,
such as four aces and a two of clubs,
four aces and a two of hearts, et cetera,
but you can see that tallying this by hand
will take forever.
This is where arrangements,
permutations, and combinations enter.
An arrangement calculation is used
when we have different versions of something,
like a machine component,
and we want to know how many ways
the components can be combined.
We simply multiply the versions together,
as in m times n.
Here's an example, an automobile dealer wants to advertise
that for $29,999 you can buy a convertible,
a two-door sedan, or a four-door model,
with your choice of either wire wheel covers
or solid wheel covers.
How many different arrangements of models and wheel covers
can the dealer offer?
There are three car styles, two wheel types,
so there are a total of three times two, or six ways.
A common person might say six combinations,
but we are going to reserve this word combination
to mean something special statistically.
As another example, if Dell wanted to advertise
the number of different laptop models that are offered,
you would simply multiply the number of versions
of each component available, incidentally,
the resulting number would be pretty big.
Here is one more example.
If we roll two dice, how many different values can result?
Since each die has six possibilities,
we simply multiply six times six and end up with 36.
If we had three dice, the answer would be 216.
Earlier, we manually created this probability distribution
for throwing two dice,
knowing that we could've simply multiplied six times six
would've been a lot easier.
Also, this 36 total outcomes becomes the denominator
of two dice throwing probability calculations.
Here is a slightly different problem,
if you have four paint colors,
how many ways can you paint the walls of a room,
assuming that you can only use one color per wall?
This is somewhat like an arrangement
with diminishing options.
In this case, we have four colors that we could use
on the first wall, but once we pick a color,
we have only three colors remaining for the second wall,
then two for the third, and finally one for the last.
The answer is four times three times two times one, or 24.
Some folks who are good with math might recognize this
as four factorial, yep, and in this specific example,
the factorial equation is really handy.
How does our example change
if we had six colors for four walls?
The calculation is similar,
six times five times four times three, or 360.
But this is not quite six factorial, if it was,
we would finish two times one, but we don't.
Hmm, a little more challenging.
Mathematicians studied this and came up with an equation
and a name for the equation, the name is permutation,
and the equation is shown on this slide.
For our problem n is equal to six, and r is equal to four,
so we would have to calculate six factorial
divided by six minus four factorial.
To do factorial calculations on a calculator,
you should buy a calculator with a factorial function key.
Don't worry, we are going to do
these calculations with Excel.
In our painting example, someone might recognize
that you might pick four colors,
but you still have a choice of which wall,
such as the north wall, to begin your painting.
Does it make any difference if you begin on the north wall,
or, let's say, the south wall?
In other words, does simply rotating the colors,
changing their order,
make the room be described as being painted differently?
Well, the answer to this is it's your call,
however, the answer to this question
will change our calculations.
If we allow rotation where order is called different,
we have a permutation.
Without considering order, this becomes a combination.
The calculation for combinations always results in a value
that is equal to or less than the value for permutations.
Sometimes I like to say C is less than or equal to P,
P being alphabetically bigger than C.
Here is the equation for calculating combinations,
again, we will do this calculation using Excel.
Here is one more example of a combination.
There are 12 players on the Carolina Forest
High School basketball team.
Coach Thompson must pick five players among the 12
on the team to comprise the starting lineup.
How many different groups are possible?
There are 12 players, so n is equal to 12.
We are going to pick five for our team, so r equals five.
Does it matter how these five players are sequenced?
I don't think so, this doesn't seem to matter,
they can play in any position.
So this is a combination where there are 792 ways.
In this problem, however,
the coach is being asked to rank the players,
so once the five players are picked,
they can be ranked in many ways.
This is a case where order does apply,
so this is a permutation,
and there are 95,040 ways.
P is much larger than C.
How about picking five cards from 52?
When we pick our five cards,
does it matter how we order them in our hand?
No, I don't think so, so this is a combination.
We can use the combination equation
to calculate that there are 2,600,000 ways
that we can be dealt five cards.
Using the combination equation is certainly faster
than tallying 2,600,000 ways.
When I learned about arrangements,
combinations, and permutations,
I thought that they were just fun equations,
I never knew that they were connected
to classical probability calculations.
It didn't matter.
As an engineering students, the equations were simple,
and I scored well on exams.
The only thing I don't like about our textbook
is that our author uses these calculations
to calculate the denominator and not the numerator.
I believe that the numerator
is actually harder to calculate,
so I think most authors are just happy getting students
how to learn how to calculate the denominator.
Maybe calculating the numerator
is reserved for advanced courses in probability.
Well, let me return to our original graphic
and review what we have covered.
We have covered some basic probability concepts,
we looked at the general rule of addition
and the general rule of multiplication.
We learned about three different probability types,
and we learned how to calculate arrangements,
permutations, and combinations.
We have covered a lot.
I think that you will enjoy my Excel model
for calculating arrangements, permutations,
and combinations, however,
you are going to have to practice.
(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.
(upbeat music)