If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Intro to theoretical probability

AP.STATS:
UNC‑2 (EU)
,
UNC‑2.A (LO)
,
UNC‑2.A.1 (EK)
,
UNC‑2.A.2 (EK)
,
UNC‑2.A.3 (EK)
,
VAR‑4 (EU)
,
VAR‑4.A (LO)
,
VAR‑4.A.1 (EK)
,
VAR‑4.A.2 (EK)
,
VAR‑4.A.3 (EK)
CCSS.Math: ,
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?

  • mr pants teal style avatar for user Lyndsey Schimmelpenningh
    what does p(h) mean?
    (220 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Egor Okhterov
      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? ;)
      (60 votes)
  • male robot hal style avatar for user Malay Kedia
    At , if we assume that the coin can land on a corner and stand straight, will the probability become 1/3?
    (17 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Gabrielle Stagg
      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.
      (21 votes)
  • female robot grace style avatar for user Anoushka B.
    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?
    (13 votes)
    Default Khan Academy avatar avatar for user
    • starky seedling style avatar for user Luis
      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.
      (10 votes)
  • mr pants teal style avatar for user 💙Rohita💙
    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?
    (11 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user rorti069
    What is the difference between Chance and Probability?
    (8 votes)
    Default Khan Academy avatar avatar for user
  • eggleston blue style avatar for user Harsh
    At Sal says "How many equally likely possibilities are there". What is meant by equally likely possibility.
    Thanks in advance!!
    (4 votes)
    Default Khan Academy avatar avatar for user
    • orange juice squid orange style avatar for user Evan
      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! (:
      (9 votes)
  • leaf grey style avatar for user ( ͡° ͜ʖ ͡°)
    Are there any "fair coins" in real life? And if there are, can you give an example?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Sid
      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.
      (2 votes)
  • blobby green style avatar for user Emily Miller
    Define chance from probability
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user saintsurinsamuel10
    i am not getting this at all
    (4 votes)
    Default Khan Academy avatar avatar for user
  • aqualine ultimate style avatar for user Stevens Hayden
    Good way to learn something you don’t know
    (4 votes)
    Default Khan Academy avatar avatar for user

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.