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### Course: Statistics and probability > Unit 7

Lesson 1: Basic theoretical probability# Intro to theoretical probability

We give you an introduction to probability through the example of flipping a quarter and rolling a die. Created by Sal Khan.

## Want to join the conversation?

- what does p(h) mean?(243 votes)
**p**(h) is the ORACLE's*answer*to your question about the future. Keep reading if you want to understand what does it mean :)

First, we want to understand what**possibly**can happen in our experiment. For example, if we roll a die, there are 6 different events that can happen:

{event-1, event-2, event-3, event-4, event-5, event-6}

We cannot favor any of these events, because we can end up with any side of the die being up.

If, for example, we win $1,000,000 when the die shows 1 OR 5 on it's side, then suddenly event-1 and event-5 stop being for us an EVENT. The real event is winning $1,000,000! This BIG event makes little event-1 and event-5 indistinguishable. So we*reorganize**our**view*on the structure of the die under the influence of that problem ― winning $1,000,000.

Now we say, there is an event of**WINNING**{event-1, event-5} and event of**LOSING**{event-2, event-3, event-4, event-6}.

There is an ORACLE that knows what happend in the past and what will happen in the future. You tell him about your big event and he gives you a number. In this video this ORACLE is called**p**. You feed him our event**WINNING**and he should give some number from 0 to 1. In his moments of absolute certainty, when his third eye opens wide (that is, when he sees the future clearly), he gives us either certain 0 or a certain 1. When his third eye is half open and he wants to sleep, he will give you numbers in between 0 and 1.

Generally, the future is determined. You will either 100% WIN or 100% LOSE. The number that mighty ORACLE gives you depends on*his**understanding*of the situation. If he totally loses the power to see the future and becomes a mere mortal like us, he will always give you the number 0.5 on every question you ask him. What is the probability of you to run across a dinosaur on the street? ;)(104 votes)

- At2:38, if we assume that the coin
**can land on a corner and stand straight**, will the probability become 1/3?(28 votes)- I would assume that if you can stand the coin straight up, without flipping it, you would have a chance of landing it on it's corner, it would just be very unlikely. Therefore, it would not be 1/3, because it does not have the same probability as the heads/tails. I would assume that the probability of it landing straight up could be about as low as 11/100, just because the coin is very thin, and (assuming it is not a sphere) your faces of the coin are wider.(34 votes)

- 1. Why does a larger number of experiments bring the percentage of say getting heads closer to fifty percent?

2. How would you describe the probability of getting all tails in the flipping coin experiment?(21 votes)- Is the Law of Large Numbers:

If you flip a coin #1 time you can have:

{[H] or [T]}

If you flip repet it 2 times you can have:

{[H,H],[H,T], or [T,H], [T,T]}

Now for the important part. If you don't care about the order you could say that the event [H,T] is equal to the event [T,H] so it'd be the same as:

{[H,H], 2[H,T], [T,T]}

the probability of each event would be:

P([H,H]) = 1/4

P([H,T]) = P([T,H]) = 2/4 =1/2

P([T,T]) = 1/4

So flip the coin 100 times and you would see that there are more combinations of HEADS & TAILS that add up to 50% each than any other.(23 votes)

- Alan, Beth Carlos and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?(19 votes)
- Take this one person at a time.

If Alan gets an A, how many other people get an A?

If Beth gets an A, how many other people get an A?

If Carlos gets an A, how many other people get an A?

That should lead you to the correct answer.(15 votes)

- What is the difference between Chance and Probability?(15 votes)
- Not much, really. They display the same statistic, but chance is formatted as a percentage while probability is a fraction. So chance would be displayed as, say 25%, while probability would be displayed as 1/4.(15 votes)

- At1:42Sal says "How many equally likely possibilities are there". What is meant by equally likely possibility.

Thanks in advance!!(8 votes)- Let me break this into two parts. First I'll talk about how many possibilities there are, then I'll talk about
*equally likely*possibilities.

The number of possibilities is the number of different things that could happen in a given scenario. If you were flipping a coin, you could get either heads or tails, making two possibilities. Or if you were taking a test, you could get the following letter grades: A, B, C, D, or F. So there are five possibilities.*Equally likely*possibilities means that all of the outcomes could happen with the same probability. Say it was a warm, sunny day, and I wanted to go swimming. The weather forecast shows these possibilities: 85% chance of no rain, 10% chance of rain, 5% chance of rain with thunderstorms. There are three possibilities in this scenario, but they are**not***equally likely*possibilities. To have the outcomes be equally likely, they each have to happen just as often as each other. Coin flips have two equally likely possibilities because heads isn't more likely than tails, and tails isn't more likely than heads. Dice rolls are another example. No number on the die is more likely to be rolled than any other.

Have a good day! (:(18 votes)

- Are there any "fair coins" in real life? And if there are, can you give an example?(10 votes)
- No, a fair coin only exists in theory. Even if the sides were deemed perfectly flat and the perimeter perfectly circular, it's extremely unlikely that that would be true all the way down to the level of atoms.(4 votes)

- Define chance from probability(7 votes)
- Chance and probability essentially have the same meaning, but you will see the word "probability" more than the word "chance" in mathematics. The word "chance" is more informal than the word "probability".(9 votes)

- what does p(h) mean(5 votes)
- It basically means the probability (p) of the coin landing on heads (h).(12 votes)

- Theoretically, if I flipped a coin 1,000 times, it would land on heads 500 times and on tails 500 times. But that's just theoretical. What if I flipped a coin 1,000 times, and every single time it landed on heads? Is there a way to calculate the chances of that happening? (even if it is very small?)(6 votes)
- First, what's the probability of getting a head per flip? It is 1 / 2 = 0.5.

Now, what's the probability of getting 2 heads in 2 flip? You have to flip head for the first time, and for the second time as well, so it will be 0.5 * 0.5 = 0.25.

You can expand this to 1000 times, which is 0.5^1000.(8 votes)

## Video transcript

What I want to do in
this video is give you at least a basic
overview of probability. Probability, a word that
you've probably heard a lot of, and you are probably a
little bit familiar with it. But hopefully,
this will give you a little deeper understanding. Let's say that I have
a fair coin over here. And so when I talk
about a fair coin, I mean that it has
an equal chance of landing on one
side or another. So you can maybe view it
as the sides are equal, their weight is the
same on either side. If I flip it in
the air, it's not more likely to land on
one side or the other. It's equally likely. And so you have one
side of this coin. So this would be
the heads I guess. Try to draw George Washington. I'll assume it's a
quarter of some kind. And the other side, of
course, is the tails. So that is heads. The other side right
over there is tails. And so if I were
to ask you, what is the probability-- I'm
going to flip a coin. And I want to know what is the
probability of getting heads. And I could write
that like this-- the probability
of getting heads. And you probably, just
based on that question, have a sense of what
probability is asking. It's asking for some
type of way of getting your hands around an event
that's fundamentally random. We don't know whether
it's heads or tails, but we can start to
describe the chances of it being heads or tails. And we'll talk about different
ways of describing that. So one way to think
about it, and this is the way that
probability tends to be introduced in textbooks,
is you say, well, look, how many different, equally
likely possibilities are there? So how many equally
likely possibilities. So number of equally--
let me write equally-- of equally likely possibilities. And of the number of
equally possibilities, I care about the number that
contain my event right here. So the number of possibilities
that meet my constraint, that meet my conditions. So in the case of the
probability of figuring out heads, what is the number of
equally likely possibilities? Well, there's only
two possibilities. We're assuming that the coin
can't land on its corner and just stand straight up. We're assuming
that it lands flat. So there's two
possibilities here, two equally likely
possibilities. You could either get heads,
or you could get tails. And what's the number
of possibilities that meet my conditions? Well, there's only one,
the condition of heads. So it'll be 1/2. So one way to think about it
is the probability of getting heads is equal to 1/2. If I wanted to write
that as a percentage, we know that 1/2 is
the same thing as 50%. Now, another way to think about
or conceptualize probability that will give you
this exact same answer is to say, well, if I were to
run the experiment of flipping a coin-- so this flip, you
view this as an experiment. I know this isn't the kind of
experiment that you're used to. You know, you normally think an
experiment is doing something in chemistry or physics
or all the rest. But an experiment
is every time you do, you run this random event. So one way to think
about probability is if I were to do this
experiment, an experiment many, many, many times-- if
I were to do it 1,000 times or a million times or a billion
times or a trillion times-- and the more the better--
what percentage of those would give me
what I care about? What percentage of those
would give me heads? And so another way to think
about this 50% probability of getting heads is if I
were to run this experiment tons of times, if I were
to run this forever, an infinite number of times,
what percentage of those would be heads? You would get this 50%. And you can run that simulation. You can flip a coin. And it's actually
a fun thing to do. I encourage you to do it. If you take 100 or 200
quarters or pennies, stick them in a big
box, shake the box so you're kind of simultaneously
flipping all of the coins, and then count how many of
those are going to be heads. And you're going to see that the
larger the number that you are doing, the more
likely you're going to get something
really close to 50%. And there's always some
chance-- even if you flipped a coin a million times, there's
some super-duper small chance that you would get all tails. But the more you
do, the more likely that things are going to
trend towards 50% of them are going to be heads. Now, let's just apply
these same ideas. And while we're starting with
probability, at least kind of the basic, this is
probably an easier thing to conceptualize. But a lot of times, this is
actually a helpful one, too, this idea that if you run the
experiment many, many, many, many times, what
percentage of those trials are going to give you
what you're asking for. In this case, it was heads. Now, let's do another
very typical example when you first
learn probability. And this is the idea
of rolling a die. So here's my die
right over here. And of course, you have, you
know, the different sides of the die. So that's the 1. That's the 2. And that's the 3. And what I want to do--
and we know, of course, that there are-- and I'm
assuming this is a fair die. And so there are six equally
likely possibilities. When you roll this, you could
get a 1, a 2, a 3, a 4, a 5, or a 6. And they're all equally likely. So if I were to ask you,
what is the probability given that I'm rolling a fair
die-- so the experiment is rolling this fair die, what is
the probability of getting a 1? Well, what are the number of
equally likely possibilities? Well, I have six equally
likely possibilities. And how many of those
meet my conditions? Well, only one of them meets
my condition, that right there. So there is a 1/6
probability of rolling a 1. What is the probability
of rolling a 1 or a 6? Well, once again, there are six
equally likely possibilities for what I can get. There are now two possibilities
that meet my conditions. I could roll a 1 or
I could roll a 6. So now there are
two possibilities that meet my constraints,
my conditions. There is a 1/3 probability
of rolling a 1 or a 6. Now, what is the
probability-- and this might seem a little silly
to even ask this question, but I'll ask it just
to make it clear. What is the probability
of rolling a 2 and a 3? And I'm just talking
about one roll of the die. Well, in any roll of the die,
I can only get a 2 or a 3. I'm not talking about taking
two rolls of this die. So in this situation,
there's six possibilities, but none of these
possibilities are 2 and a 3. None of these are 2 and a 3. 2 and a 3 cannot exist. On one trial, you cannot get a 2
and a 3 in the same experiment. Getting a 2 and a 3 are
mutually exclusive events. They cannot happen
at the same time. So the probability of
this is actually 0. There's no way to roll this
normal die and all of a sudden, you get a 2 and a 3, in fact. And I don't want to confuse
you with that, because it's kind of abstract and impossible. So let's cross this
out right over here. Now, what is the probability
of getting an even number? So once again, you have six
equally likely possibilities when I roll that die. And which of these possibilities
meet my conditions, the condition of being even? Well, 2 is even, 4 is
even, and 6 is even. So 3 of the possibilities
meet my conditions, meet my constraints. So this is 1/2. If I roll a die, I
have a 1/2 chance of getting an even number.