Original subtitles

Instructor: Hello again.

In the last video, we mentioned expected values.

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, 8 out of 10 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 wanna 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 1/4 or 0.25.

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 three 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 center 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 center.

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's going to be,

so they provide us with likely intervals instead.

That is why we often hear statements like,

"Expect between three

and five feet of snow tomorrow morning,"

or "Temperatures rising up to 90 degrees on Wednesday."

In the next lecture, we are going to show you

how to make reasonable predictions about the future

using the probability frequency distribution.

See you there, and thanks for watching.