Welcome back to An Intuitive Introduction to Probability.

In this lecture, I want to show you a first exam,

that's a little more complicated.

Calculate the probabilities there within a spreadsheet,

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's look at the sum of two dice.

We now want to roll two fair dice.

And I to to sum, and those of you who like gambling, may know the game of creps,

and that's where exactly this imported, one roll two dice.

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

Yeah, one plus one is the smallest number, that's the two.

And it goes all the way to 12, six plus six is the largest sum.

So, the basic outcomes in that sum, is two, three, four,

five all the way to 12, together, our sample space capital S.

And now, we can think of events, what's the probability of an 11 and the 12?

Or 12?

Or what's the probability of a seven?

Let's now move over to a spreadsheet that I've prepared for you, where we can

look at the possible combinations and the probability of the sums that are possible.

So here, we have our spreadsheet where we want to look now at the sum of two dice.

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

numbers that the first dice may show, and here in the top row,

the six possible numbers, that's a second I may shop.

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

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

So here, one plus one is a two.

Here, three plus one is a four.

Now notice, the different numbers show up different many times.

There's only one way to obtain a two, one plus one.

But for example, there are three ways to obtain a four,

there are also three ways to obtain a ten.

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 two, three,

four, five, 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 that table?

So, two only shows up once.

The four shows up three times, just like the ten.

And notice now, the most common number is the seven.

Six, there are six combinations to roll a sum of seven.

Six plus one, five plus two, blah, blah, blah, until one plus six.

Since we have 36 combinations in the next column,

I calculated the probability of the seven numbers for you.

Six out of 36.

One-sixth is the probability that the sum will increase as seven.

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 least likely numbers.

Only one 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.

What have you seen?

The sample space has 11 elements.

And here is now a popular mistake that I see a lot from my students.

They think we have 11 possible numbers.

So, the probability of all of them is one divided by 11.

But we just saw on the spreadsheet that's not the case, why?

Because there are several ways to get to seven.

X1, five, two.

But there's only one way to get a one.

There's just a one, one or there's only one way to get a 12, six plus six.

So, here's the danger the people sometimes think.

My sample space has many elements, probabilities are one divided by them.

That's here not correct.

The assumption that all probabilities are the same, that's not correct.

So, how can we use that classic probability or

definition to obtain these probabilities?

That requires us to take a different look, or different view of the problem and

define the outcomes in the sample space different, differently.

Instead of summing them up, let's first look at the pairs.

First I, second I.

What are now the possible outcomes?

One, one.

One, two, one, three,four, four.

Four, five, four, six, five,one, all the way to six, six.

Now we see our sample space is not 11, it has 36 pairs,

number on the first die, number on the second die,

six times six 36 combinations, and they are now all equally likely.

The probability of one, one is the same as the two on the first, and

the five on the second.

And so now, here I can use the classical probability definition in the way we saw

it in the last lecture, one divided by 36.

Now, there's a little more work to think about, what's a probability of four?

Four have three ways as a sum of occurring.

First a one, then a three, first a three, then a one, or a 2 2.

So, there are now three of these pairs that sum up to four.

Three divided by 36.

And that is exactly the probability we previously solved in the spreadsheet,

when we solved those three combinations, three in 36 of one-twelfth.

Now, let's look at different way of calculating this probabilities,

notice the following when the first dice rolls, and

shows up the one, that does not affect the second dice.

It's not like the first die calls up the second die and says, dude,

I showed the one, you better don't show one, because otherwise,

the dumb humans get confused.

No, this probability of a one is still the same

as it was before I rolled the first die.

So, that means the outcome of the first die

does not affect the outcome of the second die.

This is the concept of independence or statistical independence.

More generally, two events are called Independent.

If Event A occurring, or

not occurring, does not affect the probability of Event B occurring.

Why do we like this concept?

Because now, it's very easy for

me to calculate the probability of the intersection of two events.

Because now, if A and B are independent,

the probability of A and to B, A intersected with B,

it's just probability of A times the probability of B.

And now, I can use this with my two dice,

and ask what's the probability of first a one and then a five?

Probability of one, one in six, probability over five, one in six.

One-sixth times one-sixth is one-thirty-sixth.

And so, now I can finally show you a second way of calculating

the probability of the four.

Four, again, happens if you see first a one, then a three, first a two,

then another two, first a three, then another one.

Those three basic outcomes, or

if you want to think of them as single outcome events, are disjoint.

That means I'm allowed to add the probability of one,

three plus a probability of a two, two plus a probability of a three, one.

Each of them now I can calculate my

multiplication rule one-sixth times one-sixth for the first pair.

It's the same for the second pair, it's the same for the third and last pair, so

I get one in 36, plus one in 36, plus one divided by 36, wallah, we are back

to three divided by 36, one-twelfth the number we saw on the spreadsheet.

And that already brings me to the end of this lecture.

What are the takeaways?

In that very simple example of the sum of two fair dice,

we already realized that we have to be careful in defining the sample space.

Because that later helps us or we're calculating probabilities,

or makes calculating probabilities more tricky.

I gave you first look at a concept that is very important in probability,

the independence of events, and I showed you a cool consequence of that concept,

namely the multiplication for independent events.

We will see this in action in future lectures, so

I hope you will be back for more fun with intuitive probabilities.

Thanks for your attention.

A First Look at Statistical Independence

An Intuitive Introduction to Probability

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