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Probability with counting outcomes

The probability of getting exactly 2 heads when flipping three coins. Thinking about this by visualy depicting all of the outcomes. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • purple pi purple style avatar for user minicheesedingdong
    Is there an easier way to do this without having to write down all of the outcomes? If I get a question like this on my test it might not be as simple. Thanks.. :)
    (169 votes)
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    • leaf green style avatar for user Rajottam Ganguly
      Guys, I'll try to give you a step-wise template so that you can attempt all similar problems without wasting time. If u understand the concepts, u can solve this problem in a single step.
      1. probability of getting a head in first toss is 1/2.
      2. probability of head in second toss is 1/2.
      3. probability of tail in third toss is 1/2.
      4. Hence, probability of step 1. and step 2 AND step 3 happening is 1/2 * 1/2 * 1/2. ( Since these are independent events).
      5. Now remember, these three events can occur in ANY order. Hence three events can be arranged among themselves in 3! (3 factorial) ways. ( This rule is basic. If u have doubts, check out Sal's basic permutation videos.) So the probability of all three events happening in any order is 1/2* 1/2*1/2* 3!
      6. Finally, There are 2 heads. Order of the 2 heads DO NOT MATTER. Hence, we have counted them twice. Hence we divide by 2!. ( Its better to remember this rule: divide the final probability by the number of repeating characters' factorials. Save time in exams.)
      So this becomes 1/2*1/2*1/2*3!*1/2!. That is 3/8... Good Luck.
      (58 votes)
  • mr pants teal style avatar for user chaudharyom17
    At , Sal begins to list all the possible outcomes for flipping 3 coins. Is there any way to figure out the number of outcomes without listing them or using a chart?
    (22 votes)
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  • blobby green style avatar for user Amy Jane
    How many people are needed in a room so that the probability of two people sharing the same birthday is roughly one-half? I know the answer is 23, but I do not know how.
    (15 votes)
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    • male robot hal style avatar for user Sid
      a: Possible birthday permutations: 365^n
      b: Permutations with no shared birthdays: 365! / (365-n)

      Probability of n people NOT sharing birthday:
      b / a = 365! / (365-n)! / 365^n = 365! / (365-n)! / 365^n

      365! / (365-23)! / 365^23 = 0.493

      Probability of one or more shared birthday between 23 people:
      1 - b/a = 0.507

      I think that's an acceptable way to do it, but I've been wrong before.
      (16 votes)
  • hopper cool style avatar for user Suman Dangol
    What is the difference between theoretical probability and experimental probability?
    (10 votes)
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  • piceratops seedling style avatar for user Mohamed Kamara
    How am i supposed to solve this with a tree diagram?
    (7 votes)
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  • aqualine ultimate style avatar for user Morningcalm
    Thank you for the awesome lecture :) I suddenly got a question on my mind. I guess it's about probability and possiblity? Okay. Let's say we have to find a genious student. We had total 100 students, but couldn't find the genious one. Than if there are more than 100 students, like 500 or something? Wouldn't the possibility to find the genious student can be higher? But I think this theory contradicts with the probabilitical method. Whoa I'm getting more confused as I write it.
    (6 votes)
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    • leaf blue style avatar for user Stefen
      You are on the right track. Given that there are genius students with in the population, the larger the samples size of the population you take, the greater the possibility (that is to say, it is more probable it is) that your sample will have a genius student with in it.
      So, with a smaller sample size, eg 100 students, it is less likely you will find a genius student within it than if your sample size was 500 students, but that does not mean the finding a genius student in the smaller size is impossible, it could happen, but is just less likely to happen.
      Keep it up!
      Keep thinking and asking questions!
      (4 votes)
  • leaf green style avatar for user Hello
    What if we change this to 5 coins?
    (4 votes)
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    • blobby green style avatar for user Harikesh
      See Probability is nothing but Expected Events / All Possible Events
      So here if we flip 5 coins then all possible events = 2 * 2 * 2 * 2 * 2 = 2^5 = 32
      Expected event is getting exactly two head, if you know how combination works then you may find that event of getting exact 2 heads in 5 flips is nothing but 5C2 = 10
      So Probability of getting exactly two heads in 5 flips = 10/32 OR 5/16
      (4 votes)
  • female robot amelia style avatar for user pearpie
    Why is THH and HHT considered two different possible outcomes? Are they not the same outcome?
    (5 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      When 3 coins (or any other specified number of coins) are tossed, every sequence of heads and/or tails is equally likely. Each possible sequence occurs with probability (1/2)(1/2)(1/2) = 1/8 (or more generally with probability 1/(2^n) if n coins are tossed). However, not every combination of heads and/or tails without regard to order is equally likely. So in this type of situation, order of heads and/or tails is important for computing probabilities.

      Have a blessed, wonderful day!
      (5 votes)
  • mr pink green style avatar for user Jeremy Francis
    So I get that a deck of cards may be 52!. But, as I play some other card games I was wondering how I'd work out the possible hand combinations if say my deck contains duplicate cards?:
    ( do I just minus them from n? ) i.e (n- minus duplicates)!/k_!*1/_k ?
    (3 votes)
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    • leafers ultimate style avatar for user Yoann Nouveau
      If you have a deck of n cards, there are n! ways of arranging them.

      Now if you want to figure out how many different hands you could get with 5 cards in hand, using a deck of 52 cards for instance, you'd use the "choose function" Which will come soon after in the probability section, but here is a "sneak peak". the choose function takes 2 parameters, n = how many things to choose from (here 52, all the cards) and how many do we want to pick (here 5 for our example)
      choose(52,5) = 52! / (52-5)! * 5! = 2,598,960

      To generalize it, let's say N is total of cards to choose from, and K the number of cards you'll have in your hand
      choose(n,k)= n! / (n-k)! * k!
      (4 votes)
  • blobby green style avatar for user Isaac Danuloff
    Would this be a permutation or a combination?
    (4 votes)
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Video transcript

Find the probability of flipping exactly two heads on 3 coins. So to figure out this probability, a good place to start is just to think about all of the different possible ways that we can flip 3 coins. So we could get all tails. Tails, tails, tails. We could get tails, tails, heads. We could get tails, heads, tails. We could get tails, heads, heads. We could get heads, tails, tails. We could get heads, tails, heads. We could get heads, heads, tails. And then we could get all heads. We could get all heads over here. So there are 1, 2, 3, 4, 5, 6, 7, 8 possible outcomes. 8 possible outcomes. Now how many of the outcomes involve flipping exactly 2 heads? Let's see, that's all tails. That's 1 head, 1 head. This has 2 heads right there. That's 1 head. This is 2 heads right over there. Then this is 2 heads right over here. And then this is 3 heads, so that doesn't count. So there are 3 outcomes with exactly 2 heads. So, let me spell heads properly. 2 heads. So the probability of flipping exactly 2 heads-- And the word exactly is important, because if you didn't say exactly, then maybe 3 heads, when you flip 2 heads, so we have to say exactly 2 heads. So you don't include the situation where you get 3 heads. So the probability of flipping exactly 2 heads is equal to the 3 outcomes with 2 heads divided by the 8 possible outcomes, or 3/8. So it is equal to 3/8. And we are done.