Probability is how we measure the likelihood of something random occurring.

This can be based on what we are currently

observing or on historical knowledge or experience.

For example, when you hear that there is

an accident on the route you typically drive home from work,

you will think, "It will probably take me longer to get home today."

Or, if your favorite team is winning by 30 points at half-time,

you may think they will probably win the game.

Some probabilities can be expressed more precisely than that.

For example, the probability of flipping heads on a fair coin is one half or 50%.

In probability, an event is any collection of outcomes from a probability experiment.

That is, an event could be one outcome or a combination of outcomes.

We are typically interested in determining the probability of an event.

The sample space is the collection of all the possible outcomes.

Let's take a look at those terms as they relate to tossing a die.

If we toss a die, we will end up with a number between one and six.

Each of those numbers is a possible outcome,

so our possible outcomes are one,

two, three, four, five, and six.

When we take those six outcomes together,

we have the sample space.

That is, the sample space is the set of whole numbers between one and six.

Then, there are a lot of possible events you could calculate the probability of.

For example, rolling a three,

which is just one outcome,

or rolling an odd number,

which is the outcomes one, three,

and five, or rolling a number less than or equal to two.

I'm sure you can think of others.

There are many important rules of probability that we will learn in this module.

Our first rule is that all probabilities must be between zero and one.

Something that is guaranteed to happen has a probability of one,

for example, tossing a die and getting a number less than 10 has a probability of one.

Since the sample space of outcomes for rolling

a die only has the numbers one through six,

we must get a number less than 10.

Something that cannot happen has a probability of zero.

Tossing a die and getting a number greater than 10 has a probability of zero.

This rule may seem obvious now,

but it is an important rule to remember as you work on harder probability problems.

If you have finished your calculations and have an answer

that is negative or an answer that is greater than one,

it should signal to you that you have done something wrong.

Another rule is that probability only describes what happens in the long run.

Using the example of tossing a coin,

just because we know the probability of flipping heads is one half,

it does not mean that if you flip a coin a number of times,

you will get heads exactly half of the time.

Think about flipping a coin just two times.

It isn't at all unusual to get heads both times or in either time.

However, in the long run,

as we replicate the coin toss over and over,

we expect that what we observe will be close to the true probability.

If we flip a coin 1,000 times,

we would expect fairly close to 500 heads.

This is an application of the law of large numbers.

Our third rule is that the sum of probabilities of

all possible outcomes of an event is one.

With the coin, since the two possible outcomes are heads or tails,

adding the probability of those two outcomes together must equal one.

And a quick check of that roll shows that one half plus one half does equal one.

This leads right into our last rule for this video,

which is called the complement rule or complementation rule.

The complement rule says that the probability that event A will not

occur is equal to one minus the probability that the event A will occur.

For example, we know that the probability of rolling a six on a die is one sixth.

To calculate the probability of not rolling a six,

we could calculate one minus one sixth,

which equals five sixth.

In this simple example,

it doesn't save much time to apply the complement rule,

but there will be later examples in this course where it is a very helpful rule indeed.