Welcome back to an Intuitive Introduction to Probability.

In this lecture I wanna show you the first example

that's a little more complicated.

Calculate the probabilities, they're within a spreadsheet.

I'll show you how we can easily do that.

And then I want us to take a first look at the concept

that's very important in probability called independence,

or also sometimes statistical independence.

So, let look at "The Sum of Two Dice".

We now wanna roll two fair dice.

And I care about the sum.

Those of you who like gambling may know the game of craps

and that's where this is exactly important.

You wanna roll two dice.

So, what are the possible outcomes in this sum of 2 dice?

Yeah, 1 plus 1 is the smallest number, that's the 2.

It goes all the way to 12.

6 plus 6 is the largest sum.

So, the basic outcomes in that sum is 2, 3, 4, 5 all the way to 12

together of a sample space capital S.

And now we can think of events

what's the probability of 11 and a 12.

or 12, or what's a probability of 7.

Let's now move forward to a spreadsheet

that I've prepared for you.

We can look at the possible combinations and the probability

of the sums that are possible.

So, here we are at our spreadsheet

where we want to look now at the sum of 2 dice.

So, here in the left table I have in the left column

the 6 possible numbers as the first die may show

and here in the top row the 6 possible numbers

that a second die may show.

So, 6 times 6, notice here we have 36 combinations.

And in the table I wrote for you the sum of the 2 dice.

So, here 1 + 1 is a 2, here 3 + 1 is a 4.

Now notice,

The different numbers show up different many times

there's only one way to obtain a 2 (1+1).

But for example, there are 3 ways to obtain a 4.

There are also 3 ways to obtain a 10.

And so,

this now shows that these probabilities

will be different.

We have 36 possible combinations.

So, now here on the right I looked at the 11 possible sums

2, 3, 4, 5 all the way to 12.

And in this column called Counts

I wrote down how many times does the number on the left

the sum show up in the table.

So, 2 only shows up once.

The 4 shows up 3 times.

As like the 10.

And notice now,

most common number is the 7.

Six, there are 6 combinations

to roll a sum of 7.

6 + 1, 5 + 2, bla bla bla...

until 1 + 6.

Since we have 36 combinations, in the next column I calculated

he probability of the 7 numbers for you.

Six out of 36.

1/6 is the probability that the sum will actually show a 7.

This is the number that most common.

And those of you who like to play craps, you, of course, know that.

Two and 12 are the 2 least likely numbers.

Only 1 in 36 chance for each of them.

That's less than 3%.

So, we see here, the numbers are different, but by counting

the possible combinations

and the total number of combinations we can calculate the probabilities.

This sums up the discussion of the spreadsheet now.

I will now return to the slides, summarize what we just learned

and alert you of some dangers.