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Mega millions jackpot probability

Probability of winning the Mega Millions jackpot. Created by Sal Khan.

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  • blobby green style avatar for user Martin Kess
    Since the winnings are much greater than $176 million, doesn't this mean that it would be reasonable to buy 176 million tickets, guaranteeing that you make a profit of ~$300 million?
    (90 votes)
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  • leaf blue style avatar for user Lawrence Scott
    How was the probability of being struck by a lightning calculated? Because 1/10,000 doesn't sound like a lot.
    (17 votes)
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    • male robot hal style avatar for user jakibbe
      National Weather Service did the calculation based on reports provided to them: http://www.lightningsafety.noaa.gov/medical.htm

      ODDS OF BECOMING A LIGHTNING VICTIM
      (based on averages for 2001-2010)

      Estimated U.S. population as of 2011

      310,000,000

      Annual Number of Deaths Reported

      39

      Number of Injuries Reported

      241

      Estimated number of U.S. Deaths

      40

      Estimated number of actual Injuries

      360

      Odds of being struck by lightning in a given year (reported deaths + injuries)

      1/1,000,000

      Odds of being struck by lightning in a given year (estimated total deaths + injuries)

      1/775,000

      Odds of being struck in your lifetime (Est. 80 years)

      1/10,000

      Odds you will be affected by someone being struck (Ten people affected for every one struck)

      1/1000
      (64 votes)
  • female robot grace style avatar for user Anna
    Isn't odds the same as probability?
    (17 votes)
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  • blobby green style avatar for user trammcao
    does buying more than one tickets increase (mathematically of course) a person's odds of winning. 1 vs 100 tickets. thanks
    (9 votes)
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  • primosaur ultimate style avatar for user BA Santillan
    Where can I find a video about which to choose: a lump sum or getting money over time?
    (6 votes)
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  • leaf green style avatar for user Art
    If the odds of selecting a winning lotto number is 1 : 177M and a ticket is $1. Once the jackpot is over 177M + tax + interest on loan, why not take out a loan and buy every ticket?! Betting on the chance that you don't split the winnings with anyone this is a minimal risk "investment."
    (2 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Hmm. There is an old saying, the lottery is a tax on people who didn't think they would need algebra in real life.

      Given how many tickets are sold when the lottery gets that large, there is an excellent chance of more than one winner.

      BTW, you'll notice that they make it physically impossible to play all possible numbers. Even if you somehow managed to play 10 sets of numbers a second, round the clock, non-stop, it would take you over 200 days to play all of possible combinations, though most lotteries only give you a few days to buy tickets.

      Thus, it absolutely impossible to buy every combination for the large pay-off lotteries. That is not by accident.
      (9 votes)
  • male robot hal style avatar for user Brian Havlick
    So let's say I own a surfboard-making shop, and it's a very good shop. Only one in 1000 boards is messed up. I also have a test which is 99% accurate. If my surfboard tests bad, then what is the probability that it is bad? How would I go about doing something like this?
    (2 votes)
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    • leaf blue style avatar for user dysmnemonic
      Assuming that the boards are all tested independently and that problems making boards occur independently, a board that tests as being bad is 99% likely to actually be bad - the probability should be the same as the accuracy of the test.

      It's slightly more complicated if you consider something like the probability of selling a bad board, which should be about 1 per 100,000 -- which is 1/1000 for the chance of it actually being bad, multiplied by the 1/100 chance of it wrongly passing the test. 99 bad boards from that 100,000 would be found and not sold (99/100 detected x 1/1000 bad), while around 100 good boards would test bad (1/100 wrong tests x 999/1000 good boards).
      (5 votes)
  • blobby green style avatar for user Yugioh2728
    then how come a person has been struck by lightning 7 times
    (2 votes)
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    • leaf blue style avatar for user Stefen
      Great Question.
      Just because the probability of something is very very remote does not preclude it from ever happening. The guy that was hit 7 times, do you know what he did for a living? Perhaps it was something that put him at higher risk, like needing to be out doors and exposed even if a storm happens, like a park ranger or something, and possibly he works in an area that has a higher incidence of lightning storms due to its geographic location. All these are factors to consider.
      (5 votes)
  • blobby green style avatar for user arikast
    A related question: suppose I wanted to try to win the lottery by buying all the tickets (this isn't hypothetical, there are investment groups that actually do this!). Since there's no way to sequentially buy each number (manually filling out cards is too impractical), I resort to quickpicks, which means there is no guarantee that I'm not buying the same number multiple times.

    So the question: if I set a target of obtaining 1/2 the available numbers, how much on average do I need to spend?
    (3 votes)
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  • blobby green style avatar for user Redmond Riddell
    For those curious, you can also multiply the probabilities together and get the same answer. This helped me with the intuition: 5/56 * 4/55 * 3/54 * 2/53 * 1/52 * 1/46.

    Where 5/56 is your initial probability to pick one of your white balls, 4/55 is your conditional probability to pick another one of your balls given we have removed one ball from the pool etc, until you multiply by 1/46 which is probability of picking your red ball.
    (3 votes)
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Video transcript

I've been asked to calculate the probability of winning the Mega Millions jackpot. So I thought that's what I would do this video on. So the first thing is to make sure we understand what does winning the jackpot actually mean. So there's going to be two bins of balls. One of them is going to have 56 balls in it, so 56 in one bin. And then another bin is going to have 46 balls in it. So there are 46 balls in this bin right over here. And so what they're going to do is they're going to pick 5 balls from this bin right over here. And you have to get the exact numbers of those 5 balls. It can be in any order. So let me just draw them. So it's 1 ball-- I'll shade it so it looks like a ball-- 2 balls, 3 balls, 4 balls, and 5 balls that they're going to pick. And you just have to get the numbers in any order. So this is from a bin of 56. And then you have to get the mega ball right. And then they're going to just pick one ball from there, which they call the mega ball. And obviously, this is just going to be picked-- this is going to be one of 46, so from a bin of 46. And so to figure out the probability of winning, it's essentially going to be one of all of the possibilities of numbers that you might be able to pick. So essentially, all of the combinations of the white balls times the 46 possibilities that you might get for the mega ball. So to think about the combinations for the white balls, there's a couple of ways you could do it. If you are used to thinking in combinatorics terms, it would essentially say, well, out of a set of 56 things, I am going to choose 5 of them. So this is literally, you could view this as 56, choose 5. Or if you want to think of it in more conceptual terms, the first ball I pick, there's 56 possibilities. Since we're not replacing the ball, the next ball I pick, there's going to be 55 possibilities. The ball after that, there's going to be 54 possibilities. Ball after that, there's going to be 53 possibilities. And then the ball after that, there's going to be 52 possibilities, because I've already picked 4 balls out of that. Now, this number right over here, when you multiply it out, this is a number of permutations, if I cared about order. So if I got that exact combination. But to win this, you don't have to write them down in the same order. You just have to get those numbers in any order. And so what you want to do is you want to divide this by the number of ways that five things can actually be ordered. So what you want to do is divide this by the way that five things can be ordered. And if you're ordering five things, the first of the five things can take five different positions. Then the next one will have four positions left, and then the one after that will have three positions left. The one after that will have two positions. And then the fifth one will be completely determined because you've already placed the other four, so it's going to have only one position. So when we calculate this part right over here, this will tell us all of the combinations of just the white balls. And so let's calculate that. So just the white balls, we have 56 times 55 times 54 times 53 times 52. And we're going to divide that by 5 times 4 times 3 times 2. We don't have to multiply by 1, but I'll just do that, just to show what we're doing. And then that gives us about 3.8 million. So let me actually let me put that off screen. So let me write that number down. So this comes out to 3,819,816. So that's the number of possibilities here. So just your odds of picking just the white balls right are going to be one out of this, assuming you only have one entry. And then there's 46 possibilities for the orange balls, so you're going to multiply that times 46. And so that's going to get you-- so when you multiply it times 46-- bring the calculator back. So we're going to multiply our previous answer times 46. "Ans" just means my previous answer. I get a little under 176 million. Let me write that number down. So that gives us 175,711,536. So your odds of winning it, with one entry-- because this is the number of possibilities, and you are essentially, for $1, getting one of those possibilities. Your odds of winning is going to be 1 over this. And to put this in a little bit of context, I looked it up on the internet what your odds are of actually getting struck by lightning in your lifetime. And so your odds of getting struck by lightning in your lifetime are roughly 1 in 10,000-- chance of getting struck by lightning in your lifetime. And we can roughly say your odds of getting struck by lightning twice in your lifetime, or another way of saying it is the odds of you and your best friend both independently being struck by lightning when you're not around each other, is going to be 1 in 10,000 times 1 in 10,000. And so that will get you 1 in-- and we're going to have now eight 0's-- 1, 2, 3, 4, 5, 6, 7, 8. So that gives you 1 in 100 million. So you're actually twice-- almost, this is very rough-- you're roughly twice as likely to get struck by lightning twice in your life than to win the Mega jackpot.