John wants to toss a coin to make a random decision.

However, he only has a biased coin (that is, a coin that does not have a 50:50 chance of coming up heads or tails).  Even worse, John doesn’t know the extent of the bias, and thus has no idea about the likelihood of obtaining a head or tail.

How can John toss the coin to make a 50:50 random decision?

As ever, please do NOT post your answers, but do say if you think you have solved the puzzle and how long it took. Solution on Monday.

Yesterday we had a Rip It Up caption competition.  Here was the image:

I thought that the best caption came from Dominic Twose – “I’d leave it five minutes if I were you.” Congrats Dominic, drop us an email and we will get a signed book to you.

And finally, here is a little interview with my good self revealing exactly what Rip It Up is all about…

81 comments

    1. @Dave + @Higg’s Bosun

      In the fourth paragraph, it clearly states (as it does every single week):

      As ever, please do *NOT* post your answers, but do say if you think you have solved the puzzle and how long it took. Solution on Monday.

      We really could do with a ‘Report’ button on this blog…

  1. Its only biased if you toss the coin in the air, so to avoid this, why not pull it out of your pocket with your eys closed and.just place it on the table?

    1. “Its only biased if you toss the coin in the air, so to avoid this, why not pull it out of your pocket with your eys closed and.just place it on the table?” It says you have to toss the coin.

    2. it would be a solution, if the coin is “tossed” blindly, very shallowly and very slowly so it won’t turn… heheh

      I like solutions that provide a good alternative..

      I would suggest John to look for someone to exchange this coin with another one..

      However i think i have the “intended” solution after a few seconds

    1. It is similar in my self derived version of the von Neuman solution.
      Turning the coin over gives it reverse bias.

  2. If you know that it is biased, but not which side is more likely to come up, it really doesn’t matter that the coin is biased. That is bc you can treat the assignmet of heads / tails as a random, unbiased event.

    1. Hmmm… could be… But I don’t like this discussion untill monday, so this is about enough. We’ve already had too much spoilers in the passed weeks.

    2. Yeah, I agree with this. However it would only work for one decision (which is okay for this specific problem). But if you kept doing it, the systematic bias would become apparent.

  3. I have two solutions and I think there are more. I don’t know if I have the solution Richard is looking for.

  4. I also (almost instantaneously) thought of a general approach to un-biasing throws of the coin. But it does occur to me that we don’t know *how* biased the coin is. If it has two heads (or 2 tails) my method is stuffed!

  5. There are a number of effective methods. Though from the comments above I now realise I did not think of the one Richard likely had in mind. A good lesson in one of the trickier concepts of probability.

  6. Took a few minutes more than I expected. But passed away the commute nicely. Have a good answer. That’s fair!

  7. I have a solution that works provided the coin isn’t 100% biased (e.g. double headed), but I’m obviously failing to see the simple solution that other have spotted.

  8. I think I’ve got it… looking forward to Monday’s answer though!
    And please stop posting answers, grrr…

  9. Blimey! This one’s a tough one. I’ve been thinking about this off and on for the last 6 hours and still can’t decide.

  10. I’ve seen this problem before, but the previous time it also stated p(heads) > 0 and p(tails) > 0. I’ll bet the official answer won’t work if the coin always lands heads or always lands tails, but you can still get a 50-50 outcome.

    1. How so? If there is a general solution then how could knowing the extent of the bias make it more challenging? Surely you’ve already got the answer!

    1. You’re right about that example – but I was talking about the case where there *is* a general solution.

  11. Yep, got it in a couple of minutes and, yes, you do have to toss it in the air.
    Bear caption: (Yes, I know it’s closed, I didn’t see it previously)
    We don’t get many bears in here.
    I’m not bloody surprised, I’ve just seen the price list!

    1. my brain hurts coz I jusr read yesterday’s bear baiting bulletin, I’d say there was more than one winner.
      can’t suss the puzzle yet, hoorah

  12. I didn’t know I was anonymous, mind you, not a lot of people do?
    Do I have to log in to get my name to come up?

  13. I think Richard needs a new condition at the end of these: “As ever, if you don’t wnat your enjoyment of the puzzle spoiled by sociopaths, the illiterate or the simply stupid, don’t read the comments.”

    1. Bear Sh!tting in the woods… Toilet in the background… You are on the way there… He says “I’d leave it 5 minutes if I were you”

  14. oh I feel clever, I think I’ve got a solution for monday,
    one for tuesday, 1 for wed, 1 4 th, 1 4 etc,
    oh I don’t feel quite so clever now

  15. Think I’ve for it but not sure about my logic. Oh well I will wait till Monday to find out if I was right

  16. For those who have suggested a double-headed coin: The puzzle says he “has no idea about the likelihood of obtaining a head or tail”. If it were double-headed, he’d have a VERY GOOD idea about this likelihood. The coin must be normal – head and tail faces.

  17. About 10 seconds to think of an idea, and about 20 more seconds to draw a diagram and write out the details to make sure it worked correctly. (I haven’t yet read through the rest of the thread so I don’t know if there are any clues or spoilers).

  18. I can’t think of any solutions that depend on the outcome of the coin. That is, in all my solutions, my randomness comes from somewhere else.

    No good solution yet.

  19. *SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER* SKIP OVER THIS COMMENT

    I’VE fecking got it, and I WILL post the answer because I’ve NEVER got one before. Cover it with magicians wax, and toss it so it hits the ceiling. and sticks…. Damn….. Using Magicians wax-proof gloves of course to toss it….?

    Bollocks, I thought I had it for a second.

    Wait, I do. Get a jumbo jet in a big tall tall room, put it on it’s nose, jets pointing up, turn the jets on, cover your coin with sticky back plastic and throw it (from a safe distance, and wearing goggles) into the stream of the jet. Whereby the coin would be blasted up to the roof and will stick. Then you turn the jet off really quickly so as not to take off the roof (which is of course reinforced) and send a man on a ladder up to investigate whether the coin landed heads or tails.

    I’m sure i’m on the right track.

    Ahhhhh, wait a second. I have got it this time. Flick a copper penny or tuppence up to a magnet. I KNEW i was on the right track.

    And seriously, that was my line of thinking to get to the answer.

    *SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER SPOILER*

    FINISHED

    I better be fucking right. That whole thing took me 20 mins.

    1. And in real time as I was typing. and I don’t care if it’s not the real answer, that would work without a doubt.

  20. This reminds me of a statistical trick for dealing with a sensitive question such as “Have you ever had sex with someone of the same sex?” This technique does not identify the people who honestly answer yes, but it does give you an estimate of the proportion of yes’s.

    1. My fault, Higg’s Bosun. My comment was too cryptic. It was the first thought that occurred to me when I read the question.

      Epidemiologists often have to correlate lifestyles with disease. That means that people interviewed for a study have to answer questions about their sexual habits. You can avoid bias in the answers by having them privately toss a coin and answer “Yes” if it comes up heads and answer truthfully if it comes up tails. So you don’t know — for the individual — what a “Yes” means. But you can filter out the coin effect in the data analysis. The coin tossing combined with the fact that Richard’s problem prompts you to ignore “individual” tosses linked the two problems in my mind.

      Not a very illuminating explanation, I know. I’ve been meaning to write a blog about the epidemiological technique, replete with equations, but I never got around to it.

  21. It generalizes!

    Fuelled by the consumption of American beer (left over from the 4th of July party we give to American friends living here in Vancouver) my mind stumbled across an intuitive solution last night. Alcohol (even the trace amounts found in American beer) makes time elastic: I guess it took two or three minutes. But I could not prove that my method works.
    .
    Sleep and my morning coffee helped, I wrote down a two-line proof almost immediately.

    But what if John has three options? Yes. There is a strategy of tossing the coin that gives him a one third probability of selecting each option.

    Four options? Five? Yes. Yes. It gets complicated, but it is possible to use the biased coin to select one of n options with a probability of 1/n.

    Time for another coffee. Now what’s next? Proving Goldbach’s conjecture? Understanding the Higgs boson? Demolishing creationism and the Intelligent Design movement? No. Delete the last one: pseudoscience will always be with us.

  22. I have a solution but I’m not sure how adequate it is – it requires an undetermined amount of coin flips, potentially infinite if the coin is 100% biased.

  23. I’ve got a solution that I’m pretty sure about, but I’m sure there’s many other ways of doing this. Will be keen to see what the most efficient is on Monday.

  24. as it is now monday i feel confident in posting a possible answer. simply throw the coin four times. count the number of heads or tails. if more heads than tails then heads was the result otherwise tails. if both the same (two heads and two tails in any order) you must throw again once more.

    i have tried this many times with a coin and it produces an equal result.

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