Original subtitles

Instructor: 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're 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 four

on a six-sided die, or running a mile in under six 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 we 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 zero and one.

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 one expresses absolute certainty

of the event occurring and a probability of zero

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've a probability other than zero and one.

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

and that's what this course is going to teach you.

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 favorable.

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 two 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 four.

Once again, we have a single preferred outcome

but this time we have a greater number

of total possible outcomes,

six, therefore, the probability of this event

would look as follows, P of rolling four equals one sixth

or approximately 0.167.

Great, events can be simple or a bit more complex.

For example, what if we wanted to roll

a number divisible by three?

That means we need to get either a three or a six,

so the number of preferred outcomes becomes two.

However, the total number of possible outcomes stays

the same since the die still has six sides.

Therefore, we conclude that the probability

of rolling a number divisible by three equals two

over six which is approximately .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 the 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.

In a later lecture, we are going to define what we mean

by independent, but for now, let's observe some

more examples of probability.

What about the probability of winning the US lottery?

Even though it sounds

like something that is completely different

it actually follows the same idea.

You take the number of preferred outcomes

and divide it by all outcomes.

Now, the number of preferred outcomes we have would be equal

to the amount of different tickets we bought.

The total number of possible outcomes

on the other hand is just something we will learn

how to calculate less than an hour from now.

For the moment, just assume that there exists upward

of 175 million outcomes for the US lottery.

Therefore, each individual ticket only has a probability

of winning equal to one

over 175 million or approximately 0.000000005.

How would your chances improve if you bought two tickets?

How about five?

I don't know about you

but I like my odds of flipping a coin a lot more.

Now that you know what probabilities are

some of you might be wondering

how and when we can use them, in the next video,

we are gonna do that by introducing expected values.

Thanks for watching.