Original subtitles

Narrator: Hello, again.

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 two standard six-sided dice

and adding up the numbers on top.

We have six options for what the result

of the first one could be.

Regardless of the number we roll,

we still have six different possibilities

for what we can roll on the second die.

That gives us a total of six times six

equals 36 different outcomes for the two rolls.

For clarity, we can write out the results

in a six by six 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 seven occurs six times in the table.

This means we have six favorable outcomes.

As we already mentioned, there are 36 possible outcomes,

so the chance of getting a seven

equals six over 36, or just 1/6.

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 seven,

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 seven.

But how important is this value

if the probability associated with it is only 1/6?

The sum being equal to seven

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 seven.

Moreover, even though we are suggesting seven

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 seven

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 eight

in five different cases

means that eight has a frequency of five.

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've 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 numbers

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.

See you all there, and thanks for watching.