Original subtitles

Instructor: Welcome back folks.

Before we continue to the next section of this course,

let's talk about some of the characteristics

of probabilities and events.

For starters, let's define what a compliment is.

Simply put, a compliment of an event

is everything the event is not.

As the name suggests

the compliment helps complete the rest of the sample space.

To calculate the probability of the compliment of an event

we need to set up a few things.

For starters, if we add the probabilities

of different events, we get there sum of probabilities.

Now, if we add up all the possible outcomes of an event

we should always get one.

Remember, that having a probability of one

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 the 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 one.

So what would it mean

if we have a sum of probabilities greater than one?

Recall that probability of one expresses absolute certainty.

By definition

we cannot be any sure 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.

We will learn how to deal with such issues

when we introduce Bayesian notation less than an hour

from now.

Now, another peculiar case is if we end up with a sum

of probabilities less than one,

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 wanna tell you

that all events have compliments

and we denote them by adding an apostrophe.

For example, the compliment of the event A is denoted as

A'.

It is also worth noting that the compliment

of a compliment is the event itself.

So (A')' would equal A.

Now imagine if you were rolling a standard

six-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.

Compliments are often used when the event

we want to occur is satisfied by many outcomes.

For example, you wanna know the probability of rolling a 1,

2,

4, 5,

or 6,

that is the same as the probability of not rolling a three.

This concept is extremely useful

and will definitely come in handy during the next section.

We already said that the sum of the probabilities

of all possible outcomes equals one.

So you can probably guess how we calculate compliments.

The probability of the inverse equals one

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 1, 2, 4, 5

or 6 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 sixth.

Now, another way of describing getting 1, 2, 4, 5 or 6

is not getting a three.

Let us calculate the probability of not getting a three.

This is the compliment of getting a three,

so we know that the two should add up to one.

Therefore, the probability of not getting a three

equals one minus the probability of getting a three.

We know the P of three equals one sixth,

so the probability of not getting three is equal to one

minus one sixth.

Therefore, the probability of not getting three

is five sixths.

This shows that the probability of getting 1, 2, 4, 5

or 6 is equal to the probability of not getting a three.

Now that we have explained the basic probability notions

let us get back to the lottery example we explored

in the first lesson.

In fact, this will happen throughout a series of lessons

where we'll introduce you to the field of combinatorics.

We are going to talk about variations, permutations

and combinations, explaining what each of those terms means,

when to use them and how to compute them.

Keep up the good work.

See you in the next video, and thanks for watching.