Let’s say I have a bag. And in that bag–

I’m going to put some green cubes in that bag. And in particular, I’m going

to put eight green cubes. I’m also going to put

some spheres in that bag. Let’s say I’m going

to put nine spheres. And these are the green spheres. I’m also going to put some

yellow cubes in that bag. I’m going to put five of those. And I’m also going to put some

yellow spheres in this bag. And let’s say I

put seven of those. I’m going to stick

them all in this bag. And then I’m going

to shake that bag. And I’m going to pour it out. And I’m going to look

at the first object that falls out of that bag. And what I want to think

about in this video is what are the

probabilities of getting different types of objects? So for example, what

is the probability of getting a cube of any color? What is the probability

of getting a cube? Well, to think about that

we should think about what– or this is one way

to think about it– what are all of the equally

likely possibilities that might pop out of the bag? Well, we have 8 plus 9 is 17. 17 plus 5 is 22. 22 plus 7 is 29. So we have 29 objects. There are 29 objects in the bag. Did I do that right? This is 14, yup 29 objects. So let’s draw all of

the possible objects. I’ll represent it as this

big area right over here. So these are all the

possible objects. There are 29 possible objects. So there’s 29

equal possibilities for the outcome of my experiment

of seeing what pops out of the bag, assuming

that it’s equally likely for a cube or a

sphere to pop out first. And how many of them meet our

constraint of being a cube? Well, I have eight green cubes,

and I have five yellow cubes. So there are a

total of 13 cubes. So let me draw

that set of cubes. So there’s 13 cubes. We could draw it like

this– there are 13 cubes. This right here is the

set of cubes, this area. And I’m not drawing it exact. I’m approximating. It represents the

set of all the cubes. So the probability

of getting a cube is the number of events

that meet our criteria. So there’s 13

possible cubes that have an equally likely

chance of popping out, over all of the possible equally

likely events, which are 29. That includes the

cubes and the spheres. Now let’s ask a

different question. What is the probability of

getting a yellow object, either a cube or a sphere? So once again, how many things

meet our conditions here? Well, we have 5 plus 7. There’s 12 yellow

objects in the bag. So we have 29 equally

likely possibilities. I’ll do it in that same color. We have 29 equally

likely possibilities. And of those, 12

meet our criteria. So let me draw 12

right over here. I’ll do my best attempt. So let’s say it looks

something like– so the set of yellow objects. There are 12 objects

that are yellow. So the 12 that

meet our conditions are 12, over all the

possibilities– 29. So the probability

of getting a cube– 13 29ths, probability of

getting a yellow– 12 29ths. Now let’s ask something a

little bit more interesting. What is the probability

of getting a yellow cube? So I’ll put it in yellow. So we care about the color, now. So this thing is yellow. What is the probability of– or

as my son would say, “lello.” What is the probability

of getting a yellow cube? Well, there’s 29 equally

likely possibilities. And of those 29 equally likely

possibilities, 5 of those are yellow cubes, or

“lello” cubes, five of them. So the probability is 5 29ths. And where would we see

that on this Venn diagram that I’ve drawn? This Venn diagram is

just a way to visualize the different probabilities. And they become

interesting when you start thinking about

where sets overlap, or even where they

don’t overlap. So here we are

thinking about things that are members

of the set yellow. So they’re in this set,

and they are cubes. So this area right

over here– that’s the overlap of these two sets. So this area right

over here– this represents things that

are both yellow and cubes, because they are

inside both circles. So this right over here– let

me rewrite it right over here. So there’s five objects that

are both yellow and cubes. Now let’s ask– and this is

probably the most interesting thing to ask– what is

the probability of getting something that is yellow or or

a cube, a cube of any color? The probability of getting

something that is yellow or a cube of any

color– well, we still know that the denominator

here is going to be 29. These are all of the

equally likely possibilities that might jump out of the bag. But what are the possibilities

that meet our conditions? Well, one way to think about

it is, well, the probability– there’s 12 things that would

meet the yellow condition. So that would be this entire

circle right over here– 12 things that meet the

yellow condition. So this right over here is 12. This is the number of yellow. That is 12. And then to that, we can’t

just add the number of cubes, because if we add

the number of cubes, we’ve already counted these 5. These 5 are counted

as part of this 12. One way to think

about it is there are 7 yellow objects

that are not cubes. Those are the spheres. There are 5 yellow

objects that are cubes. And then there are 8

cubes that are not yellow. That’s one way to think about. So when we counted this

12– the number of yellow– we counted all of this. So we can’t just add

the number of cubes to it, because then we would

count this middle part again. So then we have to

essentially count cubes, the number of

cubes, which is 13. So the number of

cubes, and we’ll have to subtract out this

middle section right over here. Let me do this. So subtract out the middle

section right over here. So minus 5. So this is the number

of yellow cubes. It feels weird to write

the word yellow in green. The number of yellow cubes– or

another way to think about it– and you could just do

this math right here. 12 plus 13 minus 5 is 20. Did I do that right? 12 minus, yup, it’s 20. So that’s one way. You just get this is

equal to 20 over 29. But the more interesting

thing than even the answer of the probability

of getting that, is expressing this in terms

of the other probabilities that we figured out

earlier in the video. So let’s think about

this a little bit. We can rewrite this

fraction right over here. We can rewrite this as 12 over

29 plus 13 over 29 minus 5 over 29. And this was the

number of yellow over the total possibilities. So this right over here

was the probability of getting a yellow. This right over

here was the number of cubes over the

total possibilities. So this is plus the

probability of getting a cube. And this right over

here is the number of yellow cubes over

the total possibilities. So this right over here

was minus the probability of yellow, and a cube. I’m not going to

write it that way. Minus the probability

of yellow– I’ll write yellow in yellow–

yellow and a cube. And so what we’ve

just done here– and you could play

with the numbers. The numbers I just used

as an example right here to make things a

little bit concrete. But you can see this is

a generalizable thing. If we have the probability

of one condition or another condition– so let me

rewrite it– the probability– and I’ll just write it a

little bit more generally here. This gives us an

interesting idea. The probability of getting

one condition of an object being a member of set

a, or a member of set b is equal to the probability

that it is a member of set a, plus the probability that

is a member of set b, minus the probability

that is a member of both. And this is a really

useful result. I think sometimes it’s

called the addition rule of probability. But I want to show you that

it’s a completely common-sense thing. The reason why you can’t just

add these two probabilities is because they might

have some overlap. There’s a probability

of getting both. And if you just

added both of these, you would be double

counting that overlap, which we’ve already seen

earlier in this video. So you have to subtract

one version of the overlap out so you are not

double counting it. I’ll throw another

one other idea out. Sometimes, you

have possibilities that have no overlap. So let’s say this is the

set of all possibilities. And let’s say this is the

set that meets condition a and let me do this

in a different color. And let’s say that this is the

set that meets condition b. So in this situation,

there is no overlap. There’s no way– nothing is a

member of both sets, a and b. So in this situation, the

probability of a and b is 0. There is no overlap. And these type of conditions,

or these two events, are called mutually exclusive. So if events are

mutually exclusive, that means that they both

cannot happen at the same time. There’s no event that meets

both of these conditions. And if things are

mutually exclusive, then you can say the

probability of a or b is the probability of a plus

b, because this thing is 0. But if things are not

mutually exclusive, you would have to

subtract out the overlap. And probably the

best way to think about it is to

just always realize that you have to

subtract out the overlap. And obviously if something

is mutually exclusive, the probability of getting

a and b is going to be 0.

2019????

Great video as usual. Most valuable lesson I learned from it was to avoid repetition when counting overlapping sets. Since color and shape of object in this case are not disjoint sets, it is easy to make a mistake over counting. So, hasting into conclusion might be the most common mistake of calculating probabilities and counting.

sage of quick maths

"lello" lolly pops

proceeds to write the above message in greenI love how this took my 1 hour and 20 minute class and compressed it into an easier to understand 10 minute video.

Thanks sir

finally my concept is clear

CuBeS

very appratiate

Hey just wanna ask. What if P(Y or cube) has no 5 yellow and cubes

Getting help from this video still in 2019……CHEEEEEEERRRRRSSSSSSSSS!!!!!!!!

lello 🙂

You could've just write 8.