So far, we’ve been dealing

with one way of thinking about probability, and

that was the probability of A occurring is the

number of events that satisfy A over all of the

equally likely events. And this is all of the

equally likely events. And so in the case

of a fair coin, the probability of heads–

well, it’s a fair coin. So there’s two

equally likely events, and we’re saying one of

them satisfies being heads. So there’s a 1/2 chance

of you having a heads. The same thing for tails. If you took a die, and

you said the probability of getting an even number

when you roll the die. Well, there’s six

equally likely events, and there’s three even

numbers you could get. You could get 2, a 4, or a 6. So there’s three even numbers. So once again, you have a

1/2 chance of that happening. And this is a really

good model where you have equally likely

events happening. Now I’m going to change

things up a little bit. So I’m going to draw a line

here because this was just one way of thinking

about probability. Now we’re going to

introduce another one that’s more helpful when we can’t think

about equally likely events. And in particular, I’m going

to set up an unfair coin. So this right over here is

going to be my unfair coin. So that is my coin. Well, I could draw the coin. So it’s a gold coin this time. It is unfair. One side of that coin is a

little heavier than the other, even though it’s

meant to look fair. So it still has that picture

of some president or something on one side of it. So this is the head side. This is heads, and then,

obviously, on the back, you have tails. But as I mentioned,

this is an unfair coin. And I’m going to make

it interesting statement about this unfair coin and

one that really doesn’t fit into the mold that

I set up over here, and this interesting

statement is that we have more than a 50/50

chance of getting heads or more than a 50% chance

or more than a 1/2 chance of getting heads. I’m going to say that the

probability of getting heads for this coin right

over here is 60%. Or another way to

say it, it’s 0.6. Or another way to say

it, it is 6 out of 10. Or another way to

say it, it is 3/5. And this might make

intuitive sense to you and hopefully it

does a little bit, but I want you to

realize that this is fundamentally different

than what we were saying before because now we can’t say that

there are two equally likely events. There are two possible events. You can either get

heads or tails. We’re assuming that the

coin won’t fall on its edge. That’s impossible. So you’re either going

to get heads or tails, but they’re not

equally likely anymore. So we really can’t do

this kind of counting the number of events

that satisfy something over all of the possible events. In this situation, in order

to visualize the probability, we have to kind of take what’s

called a “frequentist approach” or think about it in terms

of frequency probability. And the way to conceptualize

a 60% of getting heads is to think, if we had a

super large number of trials, if we were to just flip

this coin a gazillion times, we would expect that 60% of

those would come up heads. It’s unclear how I

determined that this is 60%. Maybe I ran a

computer simulation. Maybe I know exactly all

of the physics of this, and I could completely model how

it’s going to fall every time. Or maybe I’ve actually

just run a ton of trials. I’ve flipped the coin a

million times, and I said, wow, 60% of those, 600,000

of those, came up heads. And then, we could make a

similar statement about tails. So if the probability

of heads is 60%, the probability

of tails– well, there’s only two

possibilities, heads or tails. So if I say the probability

of heads or tails, it’s going to be equal

to 1 because you’re going to get one of

those two things. You have 100% chance of

getting a heads or a tails, and these are mutually

exclusive events. You can’t have both of them. The probability

of tails is going to be 100% minus the

probability of getting heads, and this, of course, is 60%. So it’s 100% minus 60%, or

40%, or as a decimal, 0.4, or as a fraction, 4/10, or as

a simplified fraction, 2/5. So, once again,

this probability is saying– we can’t say

equally likely events. We could say that,

if we’re going to do a gazillion of

these, we would expect, as we get more and more and more

trials, more and more flips, 40% of those would be heads. Now, with that out of

the way, let’s actually do some problems with this. So let’s think about the

probability of getting heads on our first flip and

heads on our second flip. So, once again, these

are independent events. The coin has no memory. Regardless of what I

got on the first flip, I have an equal chance

of getting heads on the second flip. It doesn’t matter if I got

heads or tails on the first. So this is the probability of

heads on the first flip times the probability of heads

on the second flip, and we already know. The probability of heads on

any flip is going to be 60%. I’ll write it as a decimal. It makes the math a little

bit easier, 0.6, 0.6, and we can just multiply. I’ll do it right over here. So this is 0.6 times 0.6. Now, it’s always good

to do a reality check. One way to think about it

is I’m taking 6/10 of 6/10, so it should be a little

bit more than half of 6/10 or probably a little

bit more than 3/10. And we’ve explain

this in detail where we talk about

multiplying decimals, but we essentially just

multiply the numbers, not thinking about

the decimals at first. 6 times 6 is 36. And then you count

the number of digits we have to the right

of the decimal. We have one, two to the

right of the decimal. So we’re going to

have two to the right of the decimal in our answer. So it is 0.36, and

that makes sense. We’re taking 60% of 0.6. We’re taking 0.6 of 0.6, a

little bit more than half of 0.6. And, once again, it’s a

little bit more than 0.3. So this also makes sense. So it’s 0.36. Or another way to

think about it is there’s a 36%

probability that we get two heads in a row,

given this unfair coin. Remember, if it was a fair

coin, it would be 1/2 times 1/2, which is 1/4, which

is 25%, and it makes sense that this is more than that. Now, let’s think about a

slightly more complicated example. Let’s say the probability

of getting a tails on the first flip, getting

a heads on the second flip, and then getting a

tails– I’m going to do this in a new

color– and then getting a tails on the third flip. So this is going to be equal

to the probability of getting a tails on the first flip

because these are all independent events. If you know that you had

a tail on the first flip, that doesn’t affect

the probability of getting a heads

on the second flip. So times the probability

of getting a heads on the second flip,

and then that’s times the probability of getting

a tails on the third flip. And the probability of getting

a tails on any flip we know is 0.4. The probability of getting

a heads on any flip is 0.6, and then the probability of

getting tails on any flip is 0.4. And so, once again, we

can just multiply these. So 0.4 times 0.6. There’s actually a couple of

ways we can think about it. Well, we could literally say,

look, we’re multiplying 4 times 6 times 4, and then

we have three numbers behind the decimal point. So let’s do it that way. 4 times 6 is 24. 24 times 4 is 96. So we write a 96,

but remember, we have three numbers

behind the decimal point. So it’s one to the right

of the decimal there, one to the right of

the decimal there, one to the right

of decimal there. So three to the right. So we need three to the right

of the decimal in our answer. So one, two– we need one more

to the right of the decimal. So our answer is 0.096. Or another way to think about it

is– write an equal sign here– this is equal to a 9.6% chance. So there’s a little bit

less than 10% chance, or a little bit less

than 1 in 10 chance, of, when we flip this

coin three times, us getting exactly a

tails on the first flip, a heads on the second flip,

and a tails on the third flip.

you can multiply percents

for example> 40%*60%*40% = 96 000 %^3 and now you must convert into rigt unit.

(96 000%^3)/(100%*100%) = (96 000 %^3)/10 000 %^2 = 9,6 %

@kenufak or easier just remember that 90 percent = 0.9…

so it's 0.4*0.6*0.4 🙂

Sal, for your coin example I would suggest something like creating/imagining a 3 sided coin such that 2 of the sides are "tails" and the other side is "heads". I think that would visually help in explaining.

Is possible or is there a vid that does exactly 'x' numbers of events given an unfair coin. So getting exactly 2 using an unfair coin?

Sal you draw magnificent coins

JazakAllah

This video was very educational for my 7th grade daughter. I highly recommend this video if you are confused about probability. ENJOY!

Get to the fucking point and stop wasting time repeating the same words every time or drawing coins.

im black tell me what to do straight up hoe

I owe you my life ,,thanku

damn that coin is on point

What software do you use to write?

Sal, you are the best!

I watched just to admire Sal's coin drawing skills.

Please someone buy him a calculator 😂

. Ever hear of probability ……What comes after probably " Probably , But I don't know"

8:36 I don't understand how 0.096 is equal to 9.6?

Hi, I was wondering what program/method you use for writing and solving your problems during your recordings. Thank you for all that you do!

before i compelt the whole play list i thaaankkkk uuu so much god bless u khan academy in any field i wanted something i come here 😍

Thanks for teaching so clearly you are amazing. Thank u sir 🙂

Thanks

WoW! Great coin drawing! 😀

THANK YOU!!!!