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1000 coin flip

Posted by Chris on November 30, 2010 – 7:02 pm

1000 fair (and normal H/T) coins are flipped. At least 999 of them landed showing heads.

What is the probabiity that the other coin is a heads?

What is the probability of throwing at least 999 heads?

This post is under “Logic” and has 22 respond so far.
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22 Responds so far- Add one»

  1. 1. Eketahuna Said:

    1 in 2


  2. 2. Rohan Said:

    1. Given that atleast 999 coins out of 1000 are heads, the probability of the other one being heads is 1/1001.

    2. Probability of throwing atleast 999 heads is 1001/(2^1000)

    There is only one combination with all heads. On the other hand, if we say that only one amongst 1000 coins is a tails, we dont know which. It can be the first or second or third or…..or the 1000th. So we have 1000 combinations. Therefore, total combinations having atleat 999 heads are 1000+1.

    P.S. – Multiplying both answers gives us probability of getting all heads.

  3. 3. ftawil Said:



  4. 4. Frank Said:


    A: 50 %

    B: 0,5^999 = 1,8665e-301 in short


  5. 5. slavy Said:

    I am with Rohan here :)

  6. 6. Chris Said:

    Me too. Rohan has done it.

  7. 7. Rohan Said:

    Thank You slavy n Chris !!

  8. 8. Chris Said:

    Hi Rohan. No…. I thank you. You gave the right answers and with a pretty good explanation.

  9. 9. Brian Burke Said:

    I would disagree with rohans logic as each individual toss has in its own system a 50/50 chance of being heads or tales he is taking the whole system into account which is where this question throws alot of people out of wack expecting past results to affect current test. if you toss a coin 1000 times each time has a 50/50 chance of heads or tales this is not affected by the past tosses because each toss is a closed system.

  10. 10. Chris Said:

    Hi Brian. There are two main ways that people get this wrong. You mention one and fall for it yourself.

    The question doesn’t stipulate which (if any) of the coins shows a tail, only that at most 1 does. Rohan’s logic is almost impeccable (the only failing is that he doesn’t mention that the 1001 possibilities are equally likely – although I’m sure he knew/knows that). Rohan is fully aware of the points you make.

    The question is purely theoretical. With todays technology, the problem is hopelessly beyoynd simulation.

  11. 11. Grim Said:

    If at least 999 of 1000 (fair meaning balanced in our physical universe) coins falls heads then there arises the question of which universe the coins are in?

    If at least 999 of 1000 coins fall heads up then that suggests that the probability at that instant at that place is much higher than 50% that a single coin will fall heads up. The probability seems rather close to 1. Or since the outcome of one coin remains undecided it seems to be 999/1000 + the probability that a coin should fall heads up in that universe.

    Difficult to tell without knowing the probability function for coin flipping in said universe.

    :) Sweet dreams I am of to bed.

  12. 12. Chris Said:

    Hi Grim. LOL. Which universe are you from?

  13. 13. Grim Said:

    Hi Chris.

    Mostly in my own private one preferably with a large bottle of glendronach. lol.

  14. 14. Chris Said:

    Hi Grim. Don’t know Glendronach (Glenmorangie’s my tipple). Thanks for your post though, I was fairly sure that it was intended as humour.

  15. 15. Spencer Said:

    Hey, Chris. Think of it as 1000 individual tosses. While flipping 999 out of 1000 heads is unlikely, the probibility of flipping a head on each turn alone will happen 50% of the time.

  16. 16. Spencer Said:

    I read it as, what is the probibility that the coin will flip heads on the 1000th flip, which stands to be 50%. As for the second part of the problem (which solved under my middle-school level math), I see it to be .50^999
    The probibility of one toss to the power of how many tosses should equal the probibility.

    Aprox. 1.866527237^-301


  17. 17. Chris Said:

    Hi Spencer. The question carefully didn’t refer to a 1000th coin. It’s saying that 999 (but not which 999) or 1000 are showing heads.


  18. 18. Spencer Said:

    Ahh thanks! I rushed a tad bit there. Does anyone know why a coin flipped heads (granted that it falls undisturbed by everything but still air) will fall heads?

  19. 19. Chris Said:

    Hi Spencer. The coin landed heads up because it was flipped in such a way that after it left the flipper’s hand and interacted with the entire causal Universe it landed heads up.

  20. 20. Spencer Said:

    Hi Chris, I ment it will land tails. I had this problem in science today and had an urge to find out. What I cannot fathom is what coins have to do with 7th grade science!

  21. 21. Chris Said:

    Hi Spencer. Now I understand – I thought you were kidding.

    Ordinarily a coin that would have landed heads up, did so because of all the things that happened to it. But if you could do everything exactly the same, but say replaced moving air with still air, then that would affect the forces acting on the coin, and might have made it gain or lose a half spin (or odd number of half-spins).

    Think about it this way. Once you flipped the coin, it may spin 100 (a convenient number) times befor landing. A 0.5% difference in the initial flip force, would cause it to land on the other face.

    Maybe the school is warming you up for chaos theory. This is a branch of science which pays attention to the fact that a small differences in initial conditions of a system could lead to major differences in the final state of the system.

    The famous example suggests that a butterfly in Japan could cause a hurricance somewhere else. That particular one is known as the Butterfly Effect, and several science fiction movies have used it (but to change history through small changes in the past).

  22. 22. Chris Said:

    Hi Spencer. I’ve no idea what time zone this web site uses, but it’s now 12:47 am here in London, and we’re on Greenwich Mean Time.

    oops,, wrong blog, right person though.

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