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Probability: the basics

Explore what probability means and why it's useful.
Probability is simply how likely something is to happen.
Whenever we’re unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.
The best example for understanding probability is flipping a coin:
There are two possible outcomes—heads or tails.
What’s the probability of the coin landing on Heads? We can find out using the equation P(H)=?.You might intuitively know that the likelihood is half/half, or 50%.  But how do we work that out?  Probability =
Formula for calculating the probability of certain outcomes for an event
In this case:
Probability of a coin landing on heads
Probability of an event = (# of ways it can happen) / (total number of outcomes)
P(A) = (# of ways A can happen) / (Total number of outcomes)
Example 1
There are six different outcomes.
Different outcomes rolling a die
What’s the probability of rolling a one?
Probability formula for rolling a '1' on a die
What’s the probability of rolling a one or a six?
Probability of a 1 or a 6 outcome when rolling a die
Using the formula from above:
Probability formula applied
What’s the probability of rolling an even number (i.e., rolling a two, four or a six)?
Probability of rolling an even number? The formula and solution
Tips
  • The probability of an event can only be between 0 and 1 and can also be written as a percentage.
  • The probability of event A is often written as P(A).
  • If P(A)>P(B), then event A has a higher chance of occurring than event B.
  • If P(A)=P(B), then events A and B  are equally likely to occur.
Next step:
Practice basic probability skills on Khan Academy —try our stack of practice questions with useful hints and answers!
Khan Academy video wrapper
Intro to theoretical probabilitySee video transcript
Khan Academy video wrapper
Simple probability: yellow marbleSee video transcript

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  • piceratops seed style avatar for user Wendy Sugimura
    If two standard dice are rolled. What is the probability that the total of two dice is less than 6?
    (26 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user Rhyss
      less than 6 would not include 6 so
      [ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ]
      | ‎ ‎ 1-1‎ ‎ ‎ 2-1 ‎ ‎ 3-1 ‎ ‎ 4-1 ‎ ‎ |
      | ‎ ‎ 1-2 ‎ ‎ 2-2 ‎ ‎ 3-2 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎|
      | ‎ ‎ 1-3 ‎ ‎ 2-3 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ |
      | ‎ ‎ 1-4 ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎|
      [ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ‎ ]

      Length=10
      total =6x6

      ⁂ p()=10/36
      (12 votes)
  • blobby green style avatar for user bgljade
    A card is drawn from a standard deck of 52 cards. Find the probability that is
    a.) a heart or a face card.
    b.) a jack or an ace card
    c.) a 10 or a spade.
    (12 votes)
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    • duskpin sapling style avatar for user Avinash Athota
      I am just warning you, I don't know much about cards that much, so my numbers may be off.
      a. there are 13 heart cards and 12 face cards (aces aren't faces, right?), of which 3 are repeated, so 13+12-3 = 22/52 = 11/26
      b. there are 4 jacks and 4 aces, so 4+4 = 8/52 = 4/26 = 2/13
      c. there are 4 tens and 13 spades, and one 10 is repeated, so 4+13-1 = 16/52 = 8/26 = 4/13
      I hope that helps!
      (29 votes)
  • blobby green style avatar for user Isaac 🤠
    im hungry 🍞
    (6 votes)
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  • piceratops seedling style avatar for user Jim
    Can't you multiply the possibility(fraction) with the the same numerator or denominator to get a different but equivalent answer?
    Example: 3/4 chance times 3/3(numerator) equals 9/12. At my school, they say you can multiply fractions with the same numerator/denominator, but I haven't taken probability yet in my grade.
    (4 votes)
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  • starky sapling style avatar for user Trin
    does probability always have to be written like a fraction? How do you know when to write it as a percentage?
    (2 votes)
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    • leafers sapling style avatar for user green_ninja
      Usually, the question concerning probability should specify if they want either fractions or percentages. Here on KA, you can tell if they're asking for a percentage if you see a % sign by the answer box, while for fractions / decimals a small dialogue box will pop up after you click on the answer box telling you which form to put it in. (I've also seen them state which form to use in italics right after the question.)

      Hope this helps!😀
      (7 votes)
  • piceratops seed style avatar for user lpalmer22
    If there were 3 black dogs,4 brown dogs,and 2 white dog what would happen if You took 2 brown dogs away.
    (2 votes)
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    • piceratops sapling style avatar for user Nethra
      Um...there would be 7 dogs instead of 9. And there would only be 2 brown dogs now. Which is equal to the number of white dogs. Or is there a more complex reason to this? I don't know. Anyway I hope this helps.
      (4 votes)
  • winston default style avatar for user beatlemaniac
    Ok, I think I get it. So, would the probability of picking a yellow marble be 37.5%? I got 37.5% by turning 3/8 into a percentage. If I'm correct, this is a lot easier than I thought.
    (2 votes)
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  • blobby green style avatar for user moses.muramira22a
    what is the formula of probability
    (2 votes)
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    • blobby green style avatar for user daniella
      Classical Probability (Equally Likely Outcomes):
      To find the probability of an event happening, you divide the number of ways the event can happen by the total number of possible outcomes.

      Probability of an Event Not Occurring:
      If you want to find the probability of an event not happening, you subtract the probability of the event happening from 1.

      Probability of Independent Events:
      If two events don't affect each other (like flipping a coin twice), you multiply their individual probabilities to find the probability of both events happening.

      Probability of Mutually Exclusive Events:
      If two events can't happen at the same time (like rolling a die and getting a 1 or a 2), you add their individual probabilities to find the probability of either event happening.

      Conditional Probability:
      If you want to find the probability of one event happening given that another event has already happened, you divide the probability of both events happening by the probability of the second event happening.
      (1 vote)
  • blobby green style avatar for user Jan Register
    3 red marbles and 3 blue marbles. do not replace first marble in bag before picking again. probability that both marbles are blue
    (1 vote)
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    • cacteye blue style avatar for user Jerry Nilsson
      There are 6 marbles in total, and 3 of them are blue, so the probability that the first marble is blue is 3∕6 = 1∕2

      Given that the first marble was blue, there are now 5 marbles left in the bag and 2 of them are blue, and the probability that the second marble is blue as well is 2∕5

      So, the probability that both marbles are blue is 1∕2 ∙ 2∕5 = 1∕5
      (3 votes)
  • blobby green style avatar for user Jordania213
    The mall has a merry-go-round with 12 horses on the outside ring. If 12 people randomly choose those horses, what is the probability they are seated in alphabetical order? I've been stuck on this problem for so long and I have no clue to what is the right way to solve this problem?
    (2 votes)
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