Combinatorics in Real-Life The Lottery

Combinatorics in Real-Life The Lottery

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 one and sixty nine 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 OK.

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 sixty nine 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 sixty nine factorial over

five factorial times sixty nine minus five which is sixty four factorial.

The result is over eleven million.

Therefore we would have a lower chance than 1 in 11 million of correctly guessing the 5 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 OK.

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 are 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 zero point 0 0 0 0 0 0 0 0 3 great in this case.

We add 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 are 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.

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Видео Combinatorics in Real-Life The Lottery канала Data Science and AI Learner