Probability (part 5)

Probability of getting a certain number roll in Monopoly

Well, let's talk for the last video.

We were saying, well I have two dices,

like you know we're playing monopoly with two six sided dice

and I want to say what is the probability that I will get a 7?

So when I add up the two rows of the dice

was the probablity I get 7.

So I drew this grid here:and this grid esentially represents

all of the outcomes that I could get with the two dice

where on the top row that is the outcomes on dice 1 that

I can get a 1, a 2, a 3, a 4, a 5, or a 6

and similarly for dice two these are all of the outcomes

that I could get.

So each of these squares represents a particular outcome of both dices.

For example, this square means

that I got 6 on dice 1 and 6 on dice 2, right?

And of course, what does it mean that they added up to,

they added up to 12, right?

And we can go through all of them.

we could take some dice 1, dice 2,

well we see what they added up to?

Well, this is 2. This is 3,4,5,6,7.

And then this will be 3, let's go up it.

Let's see, this will be 3.

Then whis will be 4,5,6,7,8.

This will be, let's do all of them, 4,

and let's keep going on, 5,6,7,8,9.

This was 4 plus 1. This is 5,6,7,8,9,10.

And I think this is still an interesting pattern here, right?

This would be 6,7,8,9,10,11.

And this is 7,8,9,10,11,12.

So if I said, what is the probablity of getting a 7?

Well, that's all the squares that have a 7 item.

So let's see, that is, let me see if you can use this.

This fills to, that will be interesting.

So all the sevens, this one, this one, this one, this one,

this one, that one.

So what's the probability that I can get

actually I can try pretty neat that it will work.

What's the probablity of getting 7?

Well as we from our original definitions of the probability.

What is the total number equally, let me do it here.

Probability of 7.

What's the total number of equally probable outcomes?

We have 36 outcomes, since these are all equally probable, right?

There is 36 total out outcomes.

And so what's the probability of getting 7?

How many of these 36 outcomes resulted in the dice out of getting 7?

That's 1,2,3,4,5,6, so 6.

The probablity of getting a 7 is equal to 6 over 36 is equally 1/6.

So we could, you know, use this great useful

for the probability of getting any number.

We could say, and we could even, just by looking at this.

We see the most likely of all of the numbers you get 7, right?

And if you just look at the pattern,

cause it covers the whole diaconal, in terms of, and then you know,

the probablity of getting a 6 is equal to the probablity of getting 8,

you know, the probablity of getting 9

that is equal to the probablity of getting a 5 and so forth.

Let's do that. Let's see so 7 is the most probable

and just get some intonation on dice rows.

Let's see what's the socond line?

What's the probablity of getting 8?

8,8,8,8,8.

So how many 8s are there out of the total number?

Let's see, so the probability of getting an 8

is equal to 1,2,3,4,5, is equal to 5 over 36.

And that's also equal to probablity of getting a 6, right?

1,2,3,4,5,6, probablity of getting a 6.

So let me call this sixes as the same green,

just so we know that's 6.

So those all the sixes.

And this actually wouldn't hurt to memorize

cause when you play a noply you know you are,

you know, landing on board work for example.

Then you'd have to, you know,

actually I'll probablity do another video on what expected value

and expected cost of things like that

cause the probability costed a little of money

that will be very useful when you play a noply.

And so we can keep doing what's the probability of 5 was 1,2,3,4.

There is four out of the 36 outcomes of 5.

The probablity of 5 is 4/36, and that is equal 1/9.

And that's also the same as the probablity of getting a 9,

probability 9 is 1/9.

So that's interesting, I mean you know if you are play the crabs

or the thing monoply, you now have a sense of

what the different probabilities are of the different rows.

And you know, that's why I think a lot of games,

you know 7 is very important row,

because that is, actually the most probable of number.

For example, the probability of getting a 7 is higher

than the probability of getting a 9 or a 5, right?

Cause what's the probability of a 5 or a 9, that U is or, right?

Well, that's the probability of getting a 5

plus the probability of getting a 9

which is equal 1/9 plus 1/9 which is equal to 2/9,

or actually I was wrong,.

You see, that's why it could do a calculation.

1/6 is less than 2/9.

So this is a higher one.

But I can't see so I was wrong about that.

We can see the probability of getting a, let's see,

a 2 or 11, is less than the probability of getting a 7.

Let's calculate that. What's the probability of getting a 2?

Actually I should say a 3 or a 11.

I wanted it to be special or 6 that I wrote.

Probability of 2, there is only 1 situation

where I can get a 2, right?

So this is 1/36, 1 over 36 is 2.

And 11, that's 2 out of 36, right?

So 2 out of 36 is 1/18. Let me write this to 2/36.

So that equals 3/36. So that equals 1/12.

So the probability of getting a 2 which is this one,

or 11 is 1 out of 12 or the probability of getting,

so the probability of getting a 7 is twice than of getting a 2 or 11.

So let's just interesting out of, you know,

sometimes I don't knoe where is this going.

But I think it's interesting to analyse dice

because dice show up a lot.

And another way, although this green is

probably declears way of doing it.

Another way that I do it, if I don't have a green in front of mne.

If I say, what is the probability of getting a,

I don't know, let's say what's the probability of getting a 5.

Well, it's the probability of, let's say, this is dice 1.

And this is essencially the same thing as grid but it's

good to have mortal free work of this.

So, how can I get a 5?

If I get a 1 on dice 1, I get a 4 on dice 2.

If I get a 2, and then I need a 3.

If I get a 3, then I need a 2.

If I have a 4, then I need a 1.

And if I have a 5, no.

Those are only the stuations, right?

So we could say, what's the probability of getting, so this,

we need each of these probabilities,

and then the next one has to be this.

So there is 4 probabilities that kind of keeping this game in dice 1.

So that's the probability of getting 1?

That's 1/6. This is dice 1.

This is, you get a 2, 3. This is you get 4, right?

And so what's the probability of getting 1 on dice 1?

It's 1/6, right? They are all 1/6.

That's the probability of getting 2.

That's the probability of getting 3.

That's the probability of getting 4, right?

And so, even if a 1 on dice 1,

what's the probability of getting 4 then?

So then you know, there is 6 probabilities, and you know,

there is a tree, you can get a 5 or 6,

but those aren't count because we are out of the game.

So dice 1 and then on dice 2 there is one out of 6 chance

so I can get a 4.

Then there is, you know, about the other chance of the other number.

But this is the only situation we get a 5, right?

Similarly, on dice, this is dice 2, this column.

And then, if I get a 2,

what do I need on dice 2?

I need a 3, but to get exactly a 3, there is 1/6 chance again.

And of course, this is 5.

I have a 3 here, then there 1/6 chance that I get a 2,

which is exactly what I need.

And of course, there is a lot of the other things

that you can get that we are selecting for the 5s.

And if I had a 4, I am going to switch colors.

There is 1/6 chance that I get a 1 to get a 5, right?

So what are all the probabilities of these?

Well, this is 1/6 times ones.

So the probability of this. of getting a 1 and then a 4.

Let me clean this up. I am running out of time.

Actually, let me do it on this side.

So the probability of this event, these are mess and normal,

of this one, of getting 1 and then getting a 4,

well that's 1/36, right?

1/6, this is 1/6 times 1/6.

This is 1/6 then after that happened you could another 1/6.

That's 1/36 by similar logic.

This is 1/36, this is 1/36, and this is 1/36.

Each of these 1/36, and you think about that grid we drew,

each of these outcomes represent a square on that grid

getting a 2 and get a 3, getting a 1 and then getiing 4.

And then our total probability of getiing 5 is some of all this.

4/36 which is equal to 1/9.

So that's one that you don't have to draw a grid.

You could do a tree. You could do a little table like this

and say what are the ways that I can get a 5

and what's the probability of each of these, then some one up.

And they all work and in different times

different methods would be more useful.

I will see you in the next video.