Hey, everyone!
Life is filled with uncertain events and often we must consider the possible outcomes before
deciding.
We ask ourselves questions like “What is the chance of success?” and “What is the
probability that we fail?” to determine whether the risk is worth taking.
Many CEOs need to make huge decisions when investing in their research and development
departments or contemplating buyouts or mergers.
By using probability and statistical data, they can predict how likely each outcome is
and make the right call for their firm.
Some of you might be wondering: “What is this probability we are talking about?”.
Essentially, probability is the chance of something happening.
A more academic definition for this would be “the likelihood of an event occurring”.
The word event has a specific meaning when talking about probabilities.
Simply put, an event is a specific outcome or a combination of several outcomes.
These outcomes can be pretty much anything - getting Heads when flipping a coin, rolling
a 4 on a six-sided die or running a mile in under 6 minutes.
Take flipping a coin for example.
There isn’t only one single probability involved since there are two possible outcomes:
getting heads or getting tails.
That means we have two possible events and need to assign probabilities to each one.
When dealing with uncertain events we are seldom satisfied by simply knowing whether
an event is likely or unlikely.
Ideally, we want to be able to measure and compare probabilities in order to know which
event is relatively more likely.
To do so, we express probabilities numerically.
Even though we can express probabilities as percentages or fractions, conventionally we
write them out using real numbers between 0 and 1.
So, instead of using 20% or one fifth, we prefer 0.2.
All right!
Now let us briefly talk about interpreting these probability values.
Having a probability of ‘1’ expresses absolute certainty of the event occurring
and a probability of ‘0’ expresses absolute certainty of the event NOT occurring.
You probably figured this out, but higher probability values indicate a higher likelihood.
Okay!
As you can imagine, most events we are interested in would have a probability other than 0 and
1.
So, values like 0.2, 0.5, and 0.66 are what we generally expect to see.
Even without knowing any of this, you can tell some events are more likely than others.
For instance, your chance of winning the lottery isn’t as great as winning a coin toss.
That’s why you can think of probability as a field that is about quantifying exactly
how likely each of those events are on their own.
So how about we start right away?
Let’s get into it!
Generally, the probability of an event A occurring, denoted P of A, is equal to the number of
preferred outcomes over the total number of possible outcomes.
By preferred we mean outcomes that we want to see happen.
A different term people use for such outcomes is “favourable”.
Similarly, sample space, is a term used to depict all possible outcomes.
Going forward, we shall use the respective terms interchangeably.
We will go through several examples to ensure you understand the notion well.
Say, event A is flipping a coin and getting Heads.
In this case, Heads is our only preferred outcome.
Assuming the coin doesn’t just somehow stay in the air indefinitely, there are only 2
possible outcomes – heads or tails.
This means that our probability would be a half, so we write the following:
P of getting Heads, equals one half, which equals 0.5.
All right!
Now, imagine we have a standard six-sided die and we want to roll a 4.
Once again, we have a single preferred outcome, but this time we have a greater number of
total possible outcomes – 6.
Therefore, the probability of this event would look as follows:
P of rolling 4 equals: one sixth, or approximately 0 point one six seven.
Great!
Events can be simple, or a bit more complex.
For example, what if we wanted to roll a number divisible by 3?
That means we need to get either a 3 or a 6 so the number of preferred outcomes becomes
two.
However, the total number of possible outcomes stays the same since the die still has 6 sides.
Therefore, we conclude that the probability of rolling a number divisible by 3 equals:
2 over 6, which is approximately 0.33.
So far so good!
Note that the probability of two independent events occurring at the same time, is equal
to the product of all probabilities of the individual events.
For instance, the likelihood of getting the Ace of Spades equals the probability of getting
an Ace, times the probability of getting a spade.
Expected values represent what we expect the outcome to be if we run an experiment many
times.
To fully grasp the concept, we must first explain what an experiment is.
Okay!
Imagine we don’t know the probability of getting heads when flipping a coin.
We are going to try to estimate it ourselves, so we toss a coin several times.
After doing one flip and recording the outcome we complete a trial.
By completing multiple trials, we are conducting an experiment.
For example, if we toss a coin 20 times and record the 20 outcomes, that entire process
is a single experiment with 20 trials.
All right!
The probabilities we get after conducting experiments are called experimental probabilities,
whereas the ones we introduced earlier were theoretical or true probabilities.
Generally, when we are uncertain what the true probabilities are or how to compute them,
we like conducting experiments.
The experimental probabilities we get are not always equal to the theoretical ones but
are a good approximation.
For instance, eight out of ten times I go to my local shop, I have to wait in line.
Based on my experience, 80% of the time there will be a queue and 20% of the time, there
won’t be one.
I can try to calculate the true probability, but it would include far too many factors.
The experimental probability, on the other hand, is easy to compute and very useful.
Okay!
The formula we use to calculate experimental probabilities is similar to the formula applied
for the theoretical ones earlier in the course.
It is simply the number of successful trials divided by the total number of trials.
Now that we know what an experiment is, we are ready to dive into expected values!
The expected value of an event A, denoted E of A, is the outcome we expect to occur
when we run an experiment.
To clarify any confusion around the definition, let us examine the following example:
We want to know how many times we will get a spade if we draw a card 20 times.
We always record the value of the card and then return it to the deck before shuffling.
For an event with categorical outcomes, like suits, we calculate the expected value by
multiplying the theoretical probability of the event, P of A, by the number of trials
we carried out, n.
We’ve already seen how to compute the true probability of drawing a card from a specific
suit.
It is equal to one fourth or point twenty-five.
If we repeat this action 20 times, the expected value would equal 0.25, times 20, which equals
5.
An expected value of 5 means we expect to get a spade 5 times if we run the experiment.
However, nothing guarantees us getting a spade EXACTLY 5 times.
Realistically, we could get a spade 4 times, 6 times or even 20 times.
Now, for numerical outcomes we use a slightly different formula.
We take the value for every element in the sample space and multiply it by its probability.
Then, we add all of those up to get the expected value.
For instance, you are trying to hit a target with a bow and arrow.
The target has 3 layers, the outermost one is worth 10 points, the second one is worth
20 points and the bullseye is worth 100.
You have practiced enough to always be able to hit the target, but not so much that you
hit the centre every time.
The probability of hitting each layer is as follows: 0.5 for the outmost, 0.4 for the
second and 0.1 for the centre.
The expected value for this example would be “0.5 times 10, plus, 0.4 times 20, plus,
0.1 times 100”.
This is equal to “5, plus 8, plus 10, or 23”.
Wait!
We can never get 23 points with a single shot!
So why is it important to know what the expected value of an event is?
We can use expected values to make predictions about the future based on past data.
We frequently make predictions using intervals instead of specific values due to the uncertainty
the future brings.
Meteorologists often use these when forecasting the weather.
They do not know exactly how much snow, rain or wind there is going to be, so they provide
us with likely intervals instead.
That is why we often hear statements like “Expect between 3 and 5 feet of snow tomorrow
morning.”
or “Temperatures rising up to 90° on Wednesday.”.
So far, we have learned that the expected value is used when trying to predict future
events.
Sometimes the result of the expected value is confusing or doesn’t tell us much.
For instance, let us discuss a very famous example – throwing 2 standard 6-sided dice
and adding up the numbers on top.
We have 6 options for what the result of the first one could be.
Regardless of the number we roll, we still have 6 different possibilities for what we
can roll on the second die.
That gives us a total of 6, times 6, equals 36 different outcomes for the two rolls.
For clarity, we can write out the results in a 6 by 6 table, where we write the sum
of the two dice.
You can clearly see that we have repeating entries along the secondary diagonal and all
diagonals parallel to it.
Notice how 7 occurs 6 times in the table.
This means we have 6 favourable outcomes.
As we already mentioned, there are 36 possible outcomes, so the chance of getting a “7”
equals: six, over 36, or just one sixth.
Let’s also compute the expected value for this event.
Since we are dealing with numerical data, we should apply the same formula we used for
the archery problem from the last lecture.
To do so, we must assign an appropriate probability to each unique entry in the table.
Just like with the sum being 7, we do that based on the number of times the number features
in the table.
If we do so, we are going to get the expected value which ends up being 7.
But how important is this value if the probability associated with it is only one sixth?
The sum being equal to 7 might be the most probable answer, but it is still very unlikely
to occur.
Thus, we cannot reasonably bet on getting a sum of exactly 7.
Moreover, even though we are suggesting 7 is the most probable sum, how can you be sure?
What we can do is to create a probability frequency distribution.
Simply put, a probability frequency distribution is a collection of the probabilities for each
possible outcome – that’s how I know that 7 was the most probable sum of two dice.
Usually it is expressed with a graph or a table.
To understand what a probability frequency distribution looks like, we are going to construct
one right now.
Using the sample space table we already constructed.
For each unique sum, we record the amount of times it features in the table.
This value is known as the frequency of the outcome.
For example, getting a sum of 8 in 5 different cases, means that 8 has a frequency of 5.
Okay!
If we write out all the outcomes in ascending order and the frequency of each one, we construct
a frequency distribution table.
By examining this table, we can easily see how the frequency changes with the results.
Good job!
At this point, we have done most of the work!
The final step in getting the probability frequency distribution might be the most intuitive
one.
We need to transform the frequency of each outcome into a probability.
Knowing the size of the sample space, we can determine the true probabilities for each
outcome.
We simply divide the frequency for each possible outcome by the size of the sample space.
A collection of all the probabilities for the various outcomes is called a probability
frequency distribution.
As mentioned earlier, we can express this probability frequency distribution through
a table, or a graph.
All right!
On the graph, we see the probability frequency distribution.
The X axis depicts the different possible number of spades we can get, and the Y axis
represents the probability of getting each outcome.
When making predictions, we generally want our interval to have the highest probability.
We can see that the individual outcomes with the highest probability are the ones with
the highest bars in the graph.
Usually, the highest bars will form around the expected value.
Thus, the values around it would also be the values with the highest probability.
This suggests that if we want the interval with the highest probability, we should construct
it around the expected value.
Before we move on to the next section, we need to talk about the opposite of an event.
The term we use in probability theory is the “complement” and we are going to explain
why it is so important in the next lecture.
Let’s talk about some of the characteristics of probabilities and events.
For starters, let’s define what a complement is.
Simply put, a complement of an event is everything the event is not.
As the name suggests, the complement helps complete the rest of the sample space.
To calculate the probability of the complement of an event, we need to set up a few things.
For starters, if we add the probabilities of different events, we get their sum of probabilities.
Now, if we add up all the possible outcomes of an event, we should always get 1.
Remember that having a probability of 1 is the same as being 100% certain.
We are going to explain why this is true with several examples.
Okay!
Imagine you are tossing a coin.
When it falls, we are guaranteed to get either heads or tails.
Therefore, if we account for the sum of all probabilities of getting heads OR tails, we
have completely exhausted all possible outcomes.
We have accounted for the entire sample space, so we are 100% certain to get one of the two.
Since we are certain one of these will occur, the sum of their probabilities should be 1.
So, what would it mean if we have a sum of probabilities greater than 1?
Recall that probability of 1 expresses absolute certainty.
By definition, we cannot be any surer than being absolutely sure, so a probability of
1.5 does not make intuitive sense.
Instances where we can get such a sum of probabilities is when some of the assumed outcomes can occur
simultaneously.
This means we are double-counting some of the actual possible outcomes.
Now, another peculiar case is if we end up with a sum of probabilities less than 1.
Then we have surely not accounted for one or several possible outcomes.
Probability expresses the likelihood of an event occurring, so any probability less than
one is not guaranteed to occur.
Therefore, there must be some part of the sample space we have not yet accounted for.
Great!
Before we move on, we want to tell you that all events have complements and we denote
them by adding an apostrophe.
For example, the complement of the event “A” is denoted as “A apostrophe”.
It is also worth noting that the complement of a complement is the event itself, so “A
apostrophe, apostrophe” would equal “A”!
Now imagine you are rolling a standard 6-sided die and want to roll an even number.
The opposite of that would be NOT rolling an even number, which is the same as wanting
to roll an odd number.
Complements are often used when the event we want to occur is satisfied by many outcomes.
For example, you want to know the probability of rolling a 1, 2, 4, 5 or 6.
That is the same as the probability of NOT rolling a 3.
This concept is extremely useful!
We already said that the sum of the probabilities of all possible outcomes equals 1, so you
can probably guess how we calculate complements.
The probability of the inverse equals 1 minus the probability of the event itself.
To make sure you understand the notion well, we will look at the example we mentioned earlier.
The sum of probabilities of getting one, two, four, five or six is equal to the sum of the
separate probabilities.
The likelihood of each outcome is equal to one sixth, so the sum of their probabilities
adds up to five sixths.
Now, another way of describing getting “one, two, four, five or six” is “not getting
a three”.
Let us calculate the probability of not getting a 3.
This is the complement of getting a 3, so we know the two should add up to 1.
Therefore, the probability of not getting a 3 equals 1 minus the probability of getting
a 3.
We know that P of 3 equals one sixth, so the probability of not getting three is equal
to 1 minus one sixth.
Therefore, the probability of not getting 3 is five sixths.
This shows that the probability of getting one, two, four, five or six is equal to the
probability of not getting a three.