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Probability & combinations (2 of 2)

Making at least 3 out of 5 free throws. Created by Sal Khan.

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  • primosaur ultimate style avatar for user Michael
    I'm really struggling with this. Back in the early probability videos, Sal was always setting up the problems in terms of "desired outcomes divided by all possible outcomes." Through this whole video, I kept waiting for him to divide by 32, or 2^5, since there are 32 possible outcomes for throwing 5 free throws. But now that division step is skipped entirely. Why?
    (42 votes)
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    • primosaur seedling style avatar for user john.doe.13896
      I was struggling with this too, here is what clarified it for me:
      Suppose you throw a fair coin two times. This gives you the following outcomes and probabilities:
      P(HH) = 1/2 * 1/2 = 1/4
      P(HT) = 1/2 * 1/2 = 1/4
      P(TH) = 1/2 * 1/2 = 1/4
      P(TT) = 1/2 * 1/2 = 1/4
      To calculate P(HH) you can either calculate 1/2 * 1/2 since one throw has a chance of 1/2 of getting Heads or you can count the total number of possibilites which is 4 and since they all have the same probability P(HH) is 1 divided by 4. If you want the probability of two outcomes, for example HT & TH, you just add the probabilities of these two: 1/4+1/4 = 2/4 or again count these two and divide them by the total number of possibilities, which gives you the same result, 2/4. But what if the coin is unfair and has a chance of 80% of getting Heads and 20% of getting Tails? Here are the probabilities in this case:
      P(HH) = 0.8*0.8
      P(HT) = 0.8*0.2
      P(TH) = 0.2*0.8
      P(TT) = 0.2*0.2
      You can't just count the number of outcomes you are interested in and divide them by the total number here. P(HH), for example, which is 0.8*0.8 would give you 1/4 instead. But you can add the probabilities, and thats what sal did when he multiplied by 10, because there are 10 cases he was interested in with the same probability. In the example here, if you wanted to know P(HT or TH) you would calculate P(HT) + P(TH) = 0.8*0.2 + 0.8*0.2 = 0.8*0.2*2.
      (54 votes)
  • leaf red style avatar for user curtis.hart.tm
    Is there a more elegant way to calculate the probability of at least X events occurring? It seems like there has to be.

    For example, if I want to calculate the odds of a student passing a 100 question multiple choice test by purely guessing, is there no better way than to sum the probability of 70 successful guesses, 71 successes, 72 successes, etc up to 100?
    (22 votes)
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  • blobby green style avatar for user rubymnm
    Why couldn't the answer during just be 60% since it's the probability of getting three out of five? Why did he have to do all the work?
    (6 votes)
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    • leaf green style avatar for user MuliBoy
      just to correct your initial paragraph, we don't know the answer to the question "what is the proportion of baskets he will make out of 5 free throws?" - there is not telling that it will be 3 out of 5 (60%).
      But it is possible to ask "What is the most likely number of baskets he'll make out of 5?" and we can compare the different options, and we see in the video that it's most likely to be 4 baskets, because the probability for that is the highest relative to the rest, with almost 41% chance.
      (1 vote)
  • blobby green style avatar for user Vikram Raju
    This is regarding two ways of looking at the reasoning:
    1) Choosing 3 baskets out of 5, and in no particular order. Therefore, 5 x 4 x 3/ 3!
    2) Look at BBBMM as a spelling and he's asking how many ways can you rearrange it. So that's 5! considering each letter is unique. And then dividing by 3! x 2! to account for triple counting of B, and double counting of M.

    Do tell me what you (all) think. Thanks.
    (4 votes)
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  • duskpin ultimate style avatar for user Joshua Xiao
    at , Salman finds the number of combinations of getting 3 of 5 shots made, but I don't understand why he's doing this. He says there are 10 different combinations but arent MMMBB and MBMBM the same in terms of combinations?
    (6 votes)
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    • mr pink red style avatar for user andrewp18
      Good question!
      The difference between combinations and permutations is harder to see in this problem. When you consider a string of characters (a word), the strings MMMBB and MBMBM would be considered equivalent combinations but distinct permutations. However, you shouldn't view the shots as a string.

      Instead, think of it this way. I have shots 1, 2, 3, 4, and 5. Out of these 5 shots, I have to choose 3 that will be good. In this case, what we're really looking for is how to choose 3 numbers out of these 5 numbers. The correct interpretation in this problem is that 135 and 513 are equivalent combinations but distinct permutations. In either case, the person is making shot 1, shot 3, and shot 5. So what we're really doing is counting the number of ways we can choose a "group" or "committee" of 3 numbers from 5 numbers. This is simply:
      ₅𝐶₃ = 10
      You can think of this as a "string" or "word" but not in the way you suggested. The given fact is that our 5-letter word will have 3 indistinguishable M's and 2 indistinguishable B's. The question is how many distinct permutations can I make? Notice here that we have to view this as a permutation problem when we think of it as a word. In particular, we have 5! ways to arrange the letters if they were all distinguishable. But for each one of these arrangements, there are 3! ways we arrange the M's and 2! ways we arrange the B's so we have over-counted by 3! • 2!. So we divide our preliminary count by this:
      5!/(3! • 2!) = 10
      Notice that this is exactly the same quantity (and expression) as ₅𝐶₃.
      Comment if you have questions!
      (7 votes)
  • leaf green style avatar for user jim.pavlik
    I have a question that I think has kind of been asked below, but I'm still unable to wrap my mind around the answer. If the shooter makes 80% of his baskets, then why does he only have a 41% shot of making any combination of 80% of 5 baskets? It seems to me that the probability of making 80% of 5 baskets (4/5) should be pretty close, if not exactly 100%. Bear in mind, I understand the math, I'm trying to understand this conceptually.
    (5 votes)
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    • leaf green style avatar for user jim.pavlik
      I built this formula in Excel so I could manipulate some of the variables. I thought that maybe the problem would be answered by the Law of Large Numbers, i.e., that the error was in the small "sample size" and the percentage of getting 80% of baskets would approach 100% as the number of attempts went up. Turns out the opposite is true. Now I think that maybe the answer lies in the "exactly" that so many people above had had problems with. So for example, if you add up the shooters chance of getting 3, 4, and 5 baskets, you get about 71%. So now I'm wondering if this formula doesn't "build in" the probability that the shooter will get something near his 80% average. So that if we had the shooter go for 20 baskets, there's only 22% chance he will make exactly 16/20 baskets, but there's an 84% probability he will make 14, 15, 16, 17, or 18 baskets. I'm I on the right path here?
      (4 votes)
  • spunky sam blue style avatar for user Fabio Martins
    Why is it shown on this video that is more likely for someone (whose probability of making it is 80% for each FT) to make it in 4 out of 5 FT (40.96%) than to make it in 3 out of 5 (20.48%)? Shouldn't it get less likely to make it in more FT?
    (3 votes)
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    • leaf blue style avatar for user Dr C
      80% of 5 is 4.

      Therefore, we would "expect" the person to make 4 of 5 shots, its the most likely exact number. Getting exactly 4 shots is more likely than getting exactly 3 shots.

      You're probably thinking more in terms of "making at least x shots", in which case the more shots that must be made, the lower the probability. It's important to keep these two notions distinct - probability at an exact value vs over a range of values.
      (12 votes)
  • blobby green style avatar for user vsharma.2053
    Hi Sal,

    From to ,I want to know why are we even considering 5C3 because if we are taking probability of 3 successes in 5 throws then shouldn't (0.8x0.8x0.8)(0.2x0.2) be sufficient alone as the answer?Why are we considering arrangement of getting 3 successes in 5 throws which is 5C3 as combinations shouldnt matter?Please explain because the questions is asking us the probability of 3 successes in 5 throws and it doesnt say "IN HOW MANY WAYS" can we get 3 successes in 5 throws.
    (4 votes)
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    • female robot grace style avatar for user loumast17
      So it is asking the chance of making 3 shots out of 5. .8*.8*.8*.2*.2 is just one variation.

      If you think of a list of all possible outcomes of making 5 shots, there are 32 total possibilities. It might help to think of the question as asking, of those 32 outcomes, what is the probability of getting the ones that have 3 successes and 2 failures. then, every outcome has a percent chance of happening, so you would add the relevant percents up.

      Let's try it with 3 shots that have 2 successes. with 3 shots there are 8 possibilities. I am going to list them all and their probability of happening.

      BBB = .8^3
      BBM = .8^2 * .2
      BMB = .8^2 * 2
      BMM = .8 * .2^2
      MBB - .8^2 * .2
      MBM = .8 * .2^2
      MMB = .8 * .2^2
      MMM = .2^3

      If you added up all the probabilities you should get 1, which shows, or at least helps to show these are all possibilities.

      Now, if you wanted to choose all with two baskets that includes BBM, BMB, and MBB. so that's 3 of the 8 possibilities, but each one has a .8^2 * .2 percent chance of happening.

      I really hope this made sense, if not let me know.
      (6 votes)
  • blobby green style avatar for user David Rosenberg
    hello can u help me with a q
    a man stands infront of a fireing squed there are 5 shuters each have 0.5 chance of hiting what
    the chance of the person to get hit
    thank you
    (3 votes)
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  • blobby green style avatar for user Shedane.Shaw
    What is the different between at least and exactly? What if an at least question does not give you the probability of the event happening in the first place?
    (2 votes)
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    • male robot hal style avatar for user Jesse
      I'm not sure I understand all of your question, but when a problem includes "at least" then you should take into account the given value and every possible value greater than that. If a problem says "exactly", you should only consider that value.
      (3 votes)

Video transcript

Welcome back. I actually recorded this video earlier today, but then I realized my microphone wasn't plugged in. And I won't name names in terms of who unplugged it. But anyway, back to probability. My wife is giggling mischievously. Anyway, so let's do a slightly harder problem than we did before. We were dealing with fair coins, let's deal with a slightly unfair coin. Let's say I have a coin and it's-- actually instead of unfair coin let's do basketball. Let's say I'm shooting free throws and I have a free throw percentage of 80%. So when I shot a free throw, 8 out of 10 times, or 80% of the time I will make it. But that also says that 20% of time I will miss it. So given that, if I were to take-- I don't know-- 5 free throws, what is the probability that I make at least 3 of the 5 free throws? Well, let's think of it this way, what is the probability of any particular combination of making 3 out of the 5? So what do I mean by that? Let me pick a particular combination. Let's say it's a basket, basket, basket, and then I miss, miss. So that would be I made 3 out of the 5. It could be-- I don't know-- basket, miss, basket, miss, basket. And there's a bunch of them and we'll actually try to figure out how many of them there are. But what is the probability of this particular combination? Well, I have an 80% chance of making this first basket. Times 80%. That's a times right there. Times 80%. And then what's my probability of missing? Well, that's 20%, right? Times 0.2. times 0.2. So this sequels 0.8 to the third power times 0.2 squared. What's the probability of getting this exact combination? Well, it's 0.8 times-- then I miss. There's a 20% chance of that. So times 0.2 times 0.8 times 0.2 times 0.8. We could rearrange this because when you multiply numbers it doesn't matter what order you multiply them in. So this is the same thing as 0.8 times 0.8 times 0.8 times 0.2 times 0.2. So this is also the same thing as 0.8 to the third times 0.2 squared. The probability of getting any particular combination of 3 baskets and 2 misses is going to be 0.8 to third times 0.2 squared. Now what's the total probability of getting 3 out of 5? Well, it's going to be the sum of all of these combinations. You know, I could list them all, but we hopefully now are proficient enough in combinatorics and combinations to figure out how many different ways, if we have 5 baskets and we're picking-- or we have 5 shots and we're picking 3 of them to be the ones that are basket shots. So what do I mean? So let's say my 5 shots-- you know, I've shot 1, 2, 3, 4, 5. Out of these five, I'm going to choose 3. So once again, I'm putting my hat on as the god of probability and I will choose 3 of these shots to be the ones that happen to be the ones that get made. So essentially, out of 5 I am choosing 3. 5 choose 3. And what does that equal to? That's 5 factorial over 3 factorial times 5 minus 3 factorial, so that's 2 factorial. And that equals 5 times 4 times 3 times 2 times 1 over 3 times 2 times 1 times 2 times 1. We can ignore all the 1's. Let's see. We get 3 times 2 times 1. 3 times 2 times 1. We can cancel that. This 1 we can ignore, and then this 2 and then this turns into 2. So there are 10 possible combinations. These are two of them. You know, basket, basket, basket, miss, miss. Basket, miss, basket, miss, basket. And you know, it's a good exercise for you to list the other 8 of them. But using just the binomial coefficient, and hopefully you have an intuition of why that works and I'd be happy to make more videos if you feel that that you need more explanation. But I made a couple. There are 10 combinations. So essentially, the probability of getting exactly 3 out of 5 baskets, if I am an 80% free throw shot, is going to be-- switch colors. The probability of 3 out of 5 baskets is going to be equal to the probability of each of the combinations, which is 0.8 to the third times 0.2 squared. I make 3, miss 2. And then, times the total number of combinations. Each of these has a probability of this much. And then there's 10 different arrangements that I could make. There's 10 different ways of getting 3 baskets and 2 misses. So times 10, and what does that equal to? Let me get my high-end calculator here. So let's see what that is. That is 0.8 times 0.8 times 0.8 times 0.2 times 0.2 times 10. Equals 20.48. So it's essentially, a 20.48% chance that I get exactly 3 out of 5 of the baskets. Now let's make it slightly more interesting. Let's say I don't want it as a probability of 3 out of the 5. And this is actually something that probably, people are more likely to ask. What is the probability of getting at least 3 baskets? Well, if you think about it, this is the probability. This is equal to the probability of getting 3 out of 5 baskets, plus the probability of getting exactly 4 out of 5 baskets, plus the probability of getting exactly 5 out of 5 baskets. We already figured this one out. That's 20.48%. So what's the probability of getting 4 out of 5 baskets? Well, once again, if we want exactly 4 out of 5 baskets, so an example could be-- I don't know-- miss, basket, basket, basket, basket. What's the probability of any one of the combinations where I make 4 baskets? Well, it's going to be 0.8 to the fourth times-- and then I have a 20% chance that one miss. You know, it could have been basket, miss, basket, basket, basket. That's also exactly 4, but when you multiply them, the probability of getting any one of these particular combinations is exactly this-- 0.8 to the fourth times 0.2. If I have 5 baskets, how many ways can I pick 4 of them to be the ones that I make if I'm once again the god of probability? So this is going to be 0.8 to the fourth times 0.2 times 0.2 times-- out of 5 baskets, I'm choosing 4 that I'm going to make. So this is the number of combinations where I get 4 out of the 5. So what is 5 choose 4? That's 5 factorial over 4 factorial times 1 factorial. Well, that equals just 5. You can work that out. So let's just figure this out. This is going to be 0.8 times 0.8 times 0.8-- that's 3-- times 0.8. That equals-- did I do that right? Let's see. 0.1. Wait. 0.8 times 0.8-- yeah, that's right. Times 0.2 times 5. So 40.96%. So this is 40.96%. So roughly, 41% chance that I get exactly 4 out of 5 baskets. Which is interesting because that's kind of my free throw percentage. So it's almost a little less-- you know, 2/3 shot of kind of hitting my free throw percentage on the mark on that time. And what's the probability of getting 5 out of 5 Well, there's only one way of getting 5 out of 5. You have to get all 5 of them. So this is 0.8 to the fifth power. Let me get the calculator back. So it's 0.8 times 0.-- oh, wait-- times 0.8 times 0.8 times 0.8 times 0.8 equals 0.3276. So 32.77% shot. And then we can add them all up because we want the probability of at least 3. So it's going to be that, the probability of getting 5 out of 5, plus the probability of getting 4 out of 5, which is 0.4096. Plus, the probability of getting 3 out of 5. So that's 0.2048 equals 0.94208. So 94.21-- roughly, rounding-- % chance, which makes sense. If I have an 80% free throw percentage on any one shot, I have a very high probability of getting at least 3 out of 5 when I go to the free throw line. Anyway, I'm all out of time. I'll see you in the next video.