Let’s think about

the situation where we have a completely

fair coin here. So let me draw it. I’ll assume it’s a

quarter or something. Let’s see. So this is a quarter. Let me draw my best attempt at

a profile of George Washington. Well, that’ll do. So it’s a fair coin. And we’re going to flip

it a bunch of times and figure out the

different probabilities. So let’s start with a

straightforward one. Let’s just flip it once. So with one flip

of the coin, what’s the probability

of getting heads? Well, there’s two equally

likely possibilities. And the one with heads is one

of those two equally likely possibilities, so

there’s a 1/2 chance. Same thing if we

were to ask what is the probability

of getting tails? There are two equally likely

possibilities, and one of those gives us tails, so 1/2. And this is one

thing to realize. If you take the

probabilities of heads plus the probabilities

of tails, you get 1/2 plus 1/2, which is 1. And this is generally true. The sum of the probabilities

of all of the possible events should be equal to 1. And that makes sense,

because you’re adding up all of these fractions,

and the numerator will then add up to all

of the possible events. The denominator is always

all the possible events. So you have all

the possible events over all the

possible events when you add all of these things up. Now let’s take it up a notch. Let’s figure out the

probability of– I’m going to take this

coin, and I’m going to flip it twice–

the probability of getting heads and then

getting another heads. There’s two ways

to think about it. One way is to just

think about all of the different possibilities. I could get a head on

the first flip and a head on the second flip,

head on the first flip, tail on the second flip. I could get tails on the first

flip, heads on the second flip. Or I could get

tails on both flips. So there’s four distinct,

equally likely possibilities. And one way to think about

is on the first flip, I have two possibilities. On the second flip, I have

another two possibilities. I could have heads or

tails, heads or tails. And so I have four

possibilities. For each of these possibilities,

for each of these two, I have two possibilities here. So either way, I have four

equally likely possibilities. And how many of those

meet our constraints? Well, we have it

right over here, this one right

over here– having two heads meets our constraints. And there’s only one

of those possibilities. I’ve only circled one

of the four scenarios. So there’s a 1/4 chance

of that happening. Another way you could

think about this– and this is because these are

independent events. And this is a very

important idea to understand in

probability, and we’ll also study scenarios that

are not independent. But these are

independent events. What happens in the

first flip in no way affects what happens

in the second flip. And this is actually one thing

that many people don’t realize. There’s something called

the gambler’s fallacy, where someone thinks if I

got a bunch of heads in a row, then all

of a sudden, it becomes more likely on the

next flip to get a tails. That is not the case. Every flip is an

independent event. What happened in the

past in these flips does not affect the

probabilities going forward. The fact you got a heads

on the first flip in no way affects that you got a

heads on the second flip. So if you can make

that assumption, you could say that the

probability of getting heads and heads, or

heads and then heads, is going to be the same thing

as the probability of getting heads on the first flip

times the probability of getting heads

on the second flip. And we know the probability of

getting heads on the first flip is 1/2 and the probability

of getting heads on the second flip is 1/2. And so we have 1/2

times 1/2, which is equal to 1/4, which is

exactly what we got when we tried out all of the

different scenarios, all of the equally

likely possibilities. Let’s take it up another notch. Let’s figure out

the probability– and we’ve kind of

been ignoring tails, so let’s pay some

attention to tails. The probability of

getting tails and then heads and then tails– so

this exact series of events. So I’m not saying in any

order two tails and a head. I’m saying this exact order–

the first flip is a tails, second flip is a heads, and

then third flip is a tail. So once again, these are

all independent events. The fact that I got tails

on the first flip in no way affects the

probability of getting a heads on the second flip. And that in no way

affects the probability of getting a tails

on the third flip. So because these are

independent events, we could say that’s the same

thing as the probability of getting tails on

the first flip times the probability of getting

heads on the second flip times the probability of getting

tails on the third flip. And we know these are

all independent events, so this right over here is

1/2 times 1/2 times 1/2. 1/2 times 1/2 is 1/4. 1/4 times 1/2 is equal to

1/8, so this is equal to 1/8. And we can verify it. Let’s try out all of the

different scenarios again. So you could get

heads, heads, heads. You could get

heads, heads, tails. You could get

heads, tails, heads. You could get

heads, tails, tails. You can get tails, heads, heads. This is a little

tricky sometimes. You want to make sure

you’re being exhaustive in all of the different

possibilities here. You could get

tails, heads, tails. You could get

tails, tails, heads. Or you could get

tails, tails, tails. And what we see here is

that we got exactly eight equally likely possibilities. We have eight equally

likely possibilities. And the tail, heads, tails

is exactly one of them. It is this possibility

right over here. So it is 1 of 8 equally

likely possibilities.

Reminded me of high school. Your video makes it all easy to understand!

Cheers.

@ODeathKing The answer is on KhanAcademy [dot] org/about/faq#equipment : I use Camtasia Recorder ($200) + SmoothDraw3(Free) + a Wacom Bamboo Tablet ($80) on a PC. I used to use ScreenVideoRecorder($20) and Microsoft Paint (Free).

i am so happy i found u !!!!!!!!!!!

That is the best damn quarter I've ever seen drawn on a computer.

P(5H,5T) in any order? there's a math equation in there, i just cant figure it out…

He said head so many times I'm just wondering how no one has said " that's what she said " yet . .

Thanks man aloot

1:30 he says what's the probability of getting 2 heads ? She says now you're pushing it …

what I want to know is how he writes so neat with a mouse pad????

That's the best fucking drawing I've ever seen.

Listen to what he is saying and not what you're thinking. He repeatedly says headS.

thanks

haha Justsayno200, I was thinking the same thing, great detail on a quarter!

seriously, why cant professors be as concrete as Khan? By making something easy and concrete, it helps many people when you face a harder concepts/problems there you really have to think and use those basic things to understand it. This is basic and I can understand it but every individual have to verify if this is a hard problem/concept or not, not by professors who thinks everyghing is easy… For me this is an obvious concept but I couldn't use the formula on my text book but now I can! THX

what´s the probability of getting heads hehhh giggity…

but seriously, great video… helped me out

the probability of getting head is much less than 50%

Very good Video! Help a Lot!! lol

Isn't this an example of a Bernoulli trial?

if you have 3 coins that mean 2*2*2=8

2 coins 2*2=4

awesome sal!

What about Regression toward the mean?

Yay I understand my homework!!! 👍

what software do u use

An easier way of obtaining all possible outcomes for three coin tosses is to create a tree diagram. In this way, you'll be sure to account for every possible outcome.

"probability of having two heads in a row" giggity! …. I'm twelve 😐

Isn't s,p,r game effects the next game?

How did she get 6 in a row. COME ON.

numbers make me sad

What if you were dealing with a bag of 6 marbles, one black, two white, and three striped. You are too pick one marble and replace it, then pick a second marble. Would it be possible to pick two black marbles?

So why has flipping heads 200 times in a row never before happened? Need someone to explain this because 50/50 isn't cutting it.

I cannot comprehend it being possible to flip heads 200 times a row.

GED Prep right here.

probability of 4 sons to a couple ans plzz?????

Thk u this might actually help meh with my probability math test tomorrow

can anybody subscribe to me

very helpful video thanks

Why do we "multiply" the probabilities?

And not "Add"?

i need a formula for calculating the chance of gets hits (H) at least N numbers of time in X trials each having a Y probability.

i managed to figure out the formula for getting at least one hit:

that would be (1 – 1/y) ^ x

if i'm not mistaken.

but the part about calculating more than one hit over several trials is still eluding me. help :<

Thank you so much. You make great videos!

I dont understand western education, man. It's so easy yet so advanced? I gotta flip my ass here.

There is no such thing as independent probability.

Wow this is easier than I thought.

It makes no sense

Hh

It is me or do I just wonder why I never get stuff relating to math 🤔

An unbiased coin is tossed six times in a row and four different such trials are conducted.

One trial implies six tosses of the coin. If H stands for head and T stands for tail, the

following are the observations from the four trials:

(1) HTHTHT (2) TTHHHT (3) HTTHHT (4) HHHT__ __.

Which statement describing the last two coin tosses of the fourth trial has the highest

probability of being correct?

(A) Two T will occur.

(B) One H and one T will occur.

(C) Two H will occur.

(D) One H will be followed by one T.

5:26 Sal is looking for all the possible combinations of 3 flips. I noticed something interesting. The way I do it is I start with all flips equal to one event and iterate the rightmost event first, then move to the next left flip, iterate it and then iterate all of the previous right most events in the same way. I noticed Sal doing the exact same thing and just realized this is the same as counting from 0 to 7 in binary. Number of events possible determines the numbering system. And this is quite logical because think about this: if you have 2 digits, how many ways are there to put values 0-9 into those digits? Answer is 100. You start with 00, 01,02..56,57…99. So yeah what Sal did there to count the number of events is a valid method. Sucks he didnt mention that.

Thats a nice quarter.

Usually if I get a heads its probably followed up by a tails, if you know what I mean ;;((

When would it be necessary to subtract the probabilities from 1 to find the combined probabilities?

Hiiii

This guy is such a wiz. I followed along and now understand my assignment.

So this isn’t right?

Heads : (1) 1/2

(2) 1/4

(3) 1/8

(4) 1/16

(5) 1/32

So wouldn’t the next flip have a 63/64 chance of being tails? Not a 1/2?

Question? I get why the total number of outcomes is 4 but at 2:14 he multiplies 2 times 2. Why? Why does he multiply the number of heads together?

Yeah I still failed my math test

What if instead of fractions you have lets say 72% in the first event, 54% in the second, 31% in the third?

Thank u soooo much!! I was having such a hard time understanding this

What if you want to know which is the provability to have one time heads and one time tails but without minding which one happens first and which one happens second?

Currently studying for the Praxis and struggling with this (I'm a Lib Arts graduate so math is a struggle). Whenever I turn on a video and hear this guy's voice, I'm like, "YESSSS I'm actually going to GET IT!" Hahaha