Original subtitles

Instructor: Welcome back.

In the beginning of the course

we talked about a person's chances of winning the lottery

but we didn't go through any actual calculations.

For starters, let us define the rules of this lottery.

We need to pick five numbers between 1 and 69

and a Powerball number between 1 and 26.

As we mentioned in the beginning of the course,

the likelihood of two independent events

occurring simultaneously

equals the product of their individual probabilities.

In this case one event would be

guessing the Powerball number

and the other would be getting the correct five numbers.

Okay, let's start with the simpler part,

picking our Powerball number.

Since there is only one favorable outcome,

the chance of picking the correct Powerball number

is 1/26.

Now, how about picking five numbers out of 69?

As you know, order does not matter with lottery numbers

so we would have to use combinations.

Obviously, we cannot have the same value twice.

This means that we are dealing with combinations

without repetition.

Therefore, let's apply the relevant formula.

This suggests the total number of possible ways

to pick our five numbers is 69 factorial

over five factorial

times 69 minus five,

which is 64 factorial.

The result is over 11 million.

Therefore, we would've a lower chance than one in 11 million

of correctly guessing the five numbers.

To win the grand prize

we would also have to correctly guess the Powerball number,

so guessing all the numbers becomes 26 times less likely.

Therefore there are almost 300 million

different possible outcomes.

Okay, to get the probability of winning

we need two pieces of information

number of favorable outcomes

and the number of all possible outcomes.

We already have the latter.

What about the number of favorable outcomes?

Well, they're going to be equal

to the number of tickets we buy.

Assuming we participate with a single ticket

we can calculate the probability of winning.

Using the favorable overall formula

we get approximately one over 300 million.

In other words, the probability of winning the lottery

with a single ticket is approximately equal

to a figure that is slightly above 0.000000003.

Great.

In this case, we had two events

that needed to happen concurrently

for us to win the big jackpot.

The first one consisted of choosing the five numbers

and the second one was picking the correct Powerball number.

With this notion,

we more or less exhausted the permutations,

variations and combinations topic.

In the next video, we're going to make a quick recap

of everything we covered in this section

and then we'll be ready to move forward.

Thanks for watching.