He’s looking for probability puzzles

Adrian Torchiana writes:

I recently created a little probability puzzle app for android, and I was wondering whether you have any suggestions for puzzles that are engaging, approachable to someone who hasn’t taken a probability course, and don’t involve coins or dice. I think my easy puzzles are easy enough, but I’m having trouble thinking of new ones that are different enough from what’s already there.

Feel free to post any suggestions in the comments.

40 thoughts on “He’s looking for probability puzzles

  1. What would be really cool for some of these problems would be a little simulation engine that “proves” the answer.

    For suggestions, I always liked the sock problem: Say you have x red socks and y blue socks in your drawer. If you pick z socks at random, how big does z have to be to guarantee that you have a matching colored pair? And of course you could use the Monty Hall problem.

    • Thanks again for posting. I have heard of Mosteller’s book — I should probably add a credits page to my app pointing out that most of the puzzles are borrowed from someplace, whether a book or a class I took.

    • No, it’s not. I recently saw a nice explanation of this, but can’t find the link right now. So, I’ll try to summarize.

      In the usual version of the problem, we assume Monty ALWAYS shows a non-winning door and asks if you want to switch. But that’s not it at all. Monty would sometimes do this, sometimes do something else entertaining.

      Therefore, it’s not clear whether he’s purposely trying to lead you away from the big prize (so you shouldn’t switch) or whether it’s pre-scripted that he’s always going to show you a non-winner (in which case you should switch).

      So, unless we can reasonably suppose the data generating process Monty is using, we can’t determine the answer to the problem.

      If we assume a certain data generating process, we can arrive at an answer, but that answer isn’t right unless we are right about the data generating process.

      Similarly (and with more financial implications): if we assume the investment manager who did best last year will continue to perform above average, then we should let them manage our assets. If we assume that was mostly luck and is not likely to be replicated (e.g. the efficient market hypothesis is mostly true) then we should buy index funds. In this case, there’s sufficient information to convince me which data generating process is more correct, but others disagree.

      • > that answer isn’t right unless we are right about the data generating process.
        Yes, of course.

        But the disappointing feature of the puzzle is the implicit assumption that when he can open two doors he chooses randomly. We know that’s not the case, individuals don’t do things randomly. Maybe he has a tendency to open the higher numbered door.

        Here is the disappointment caused by that.
        The implicit assumption of random choice supports the intuition that the door opened is not informative in this particular setup.
        Hence the well known _solution_.

        In the extremes of Monte always opening the higher number door or vice versa – it is very informative but not enough to make switching the wrong choice.

        So there was a lost opportunity to show that to find out if something is informative just check if conditioning on it changes the posterior.
        (Some of this, not by me, was on the wiki page at one time.)

        • When I’ve done this problem in class, I’ve always been very careful to state all of the assumptions correctly, including the “Monty chooses randomly if contestant chooses door with prize.” And, after doing the classic problem, I also discuss “Ignorant Monty,” “Monty From Hell,” and “Random Monty,” the last being a 50% mixture model of “Classic Monty” and “Monty From Hell”.

          “Ignorant Monty” just opens a door, which may have a prize or not. That Monty doesn’t know where the prize is.

          “Monty From Hell” opens the door with the prize if you’ve chosen wrong and says “Ha, ha, you guessed wrong.” Only if you guessed right does he open a door and offer you the chance to switch.

          These are good exercises for a seminar-style class as all the ramifications of the various scenarios can be discussed.

        • Actually, I call that last example “Mixture Monty,” not “Random Monty.”

          The payoffs are the same as for “Ignorant Monty”.

  2. I like one that I made up… http://www.cs.utoronto.ca/~radford/csc2506/puzzle

    For this puzzle, as for the Monty Hall one, it can be quite difficult to get people to accept the correct answer. People tend to commit to an answer early, and then be very reluctant to admit that they were wrong, resorting to arguments about how the wording is actually ambiguous, real goats don’t behave that way, etc. (see above for MH) when nobody was actually confused (until they decided they wanted to be confused).

    • Radford:

      Your puzzle is phrased in an amusingly retro way. Anyway, I tried it and it seems like the odds are obviously 2-to-1. But now I’m worried that I’m thinking about it wrong!

    • Well they certainly have one of the two – 2 boys or 2 girls. There is no reason for either to be more likely. So there is a 0.5 probability for 2 boys and a 0.5 probability for 2 girls.

    • Neat! Are you looking for this;

      * Probability of seeing a girl’s bedroom is n.girls/3.
      * Probability of seeing a letter about boys is 1/0 if there are any boys/no boys in the family.

      If so, I make it 2 to 1 in favor of 2 girls.

      • George:

        That was my reasoning too. But as commenter J. Cross points out below, if the kids go to multiple schools, the probability of seeing that letter could increase with the number of boys.

        • Yes, that’s an astute comment.

          In terms of puzzle-writing, I wondered if the letter’s wording was too obviously weird, and that this would tip off some readers that it’s where the reasoning must be careful. (Apologies if this is getting too close to “How To Solve It”. But at least attractiveness of parents isn’t mentioned…)

        • George:

          The amazing thing to me is how hard for me to solve this sort of problem just by guessing, and how trivial it is to solve it by thinking of the likelihood ratio. (Which, by the way, ignores the prior odds, which are .512/.488 in favor of the 2 boys and 1 girl, rather than 2 girls and one boy.)

    • Hi Radford,

      Does this method of trying to teach it work?

      First piece of information: they have three children.
      So we have the following possibilities:

      BBB
      BBG
      BGG
      GGG

      Upon stumbling into a room, we find a girl.
      This eliminates BBB.
      So we are left with:

      BBG with 1/3 chance of stumbling into a girl room
      BGG with 2/3 chance of stumbling into a girls room
      GGG with 3/3 chance of stumbling into a girls room

      We then learn there is at least one boy.
      This eliminates GGG.
      So we are left with:

      BBG at 1/3
      BGG at 2/3

      Is this a reasonable way to teach it?

  3. Is it safe to assume that all three kids attend the same school? If so, I agree that it’s be 2-to-1 (two girls is twice as likely as two boys). If they attend three different schools, I think it’s 1-to-1.

      • Ah! Multiple schools. From which the kids all arrive home at the same time. But did you notice that the phrase “those are the kids’ bedrooms” doesn’t specify *whose* kids those are? Could those be the bedrooms of some exchange students visiting from Chile, while the couple’s own kids have bedrooms in the basement? Then there’s the possibility of transgender kids. Perhaps they *used to* have a boy in the school, which is why they got the letter, but that boy is now a girl. The complications are endless…

        Or you could just read the last paragraph.

        • Radford:

          Yes, I agree that if they’re all coming home at the same time, this is evidence for a single school. On the other hand, 6pm is too late to be coming home from school, so it must be an after-school program. Or maybe the older kid is bringing the two younger kids home, I’m not sure.

          And, yes, I read your last paragraph. But the thing is, as in the Monty Hall problem, the assumptions that might seem obvious to you, as the poser of the problem, are not always so obvious to me, the reader of the problem. If you wanted to make it unambiguously 1 school, I guess you could rephrase the intro to the problem and have the parents say, “before our three kids come home from their school . . .”

          Also, there’s one more difference here which is that the bias goes in one direction: if the one-school assumption does not hold, it changes the probabilities in a systematic way, so it’s not quite like the example you gave in your last paragraph where the answer could change in either direction.

          To me—but I’m just one reader—the bedrooms are obviously one per kid, and each kid is a boy or a girl. When reading the problem I’d actually thought about those alternatives but they seemed too obviously tricky so I dismissed them. I hadn’t thought of the multiple-school issue at all, but once the other commenter brought it up, it did seem like a possibility.

          It’s fine, I’m not saying there’s anything wrong with the problem—if anything, it makes it a better problem when we start considering these alternative possibilities. It’s notoriously difficult to write unambiguous word problems, as I’ve learned from years of students discovering alternative interpretations of my exam questions!

        • At some point, careful elucidation of exactly those points that are crucial starts to become too much of a hint.

          For this problem, one of my goals was to convey all information in a “realistic” manner, because the manner in which the information is conveyed actually does matter. This is why some related problems like “A couple have three kids. One of them is a boy. What is the probability that they have more than one boy?” don’t work, because they fail to specify how the information that one is a boy was obtained, which is actually crucial.

          Seeing as it *is* a puzzle, I do try to distract the reader by summarizing the information about boys and girls symmetrically, even though the information about boys and about girls was not obtained in a symmetrical manner.

        • Radford:

          Yes, I agree. It’s surprisingly hard to convey a likelihood function in a word problem, in part because of a fundamental difficulty of statistics, which is that it is necessary to specify the distribution for things that have not occurred (or, to put it another way, to determine normalizing factors). I think this is a place where our intuition fails us. It is perhaps similar to the well-known problem where you have to figure out which cards to turn over to test a deterministic hypothesis, and people tend to forget about the contrapositive.

        • Andrew:

          Constructing an adequate probability model for how the data came about, likely is always challenging even with well designed and executed studies (just recasting your point to better grasp it.)

          But like prior models, they never should be doubted or checked ;-)

        • But perhaps like a grammatically correct nonsense sentence that no matter how careful constructed can not prevent meaningful interpretation by someone – setting a puzzle that does not involve assumptions folks find unclear/mistaken – simply can’t be done.

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