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  • #91
    Re: Monty Hall, we have a PROBLEM

    I remember reading this story about a computer programming tournament to write a program that could win at rock-paper-scissors.

    It was a round-robin format, each program would play a bunch of games against each of the other programs, aiming for the most overall wins. There was all kind of debate on what strategies would be the most effective.

    One team got clever and simply entered several dozen identical programs, save for one feature. Each program had a distinct sequence of 5 moves it would throw to open each game, and they were programmed to recognize the other programs' opening sequences. If two of their programs met and recognized each other's sequences, one would immediately begin throwing nothing but rock and the other nothing but paper.

    They took something like the top 10 spots AND the bottom 10 spots. Next year they limited entries to one per team
    Michigan Tech Huskies Pep Band: There's No Use Trying To Talk. No Human Sound Can Stand Up To This. Loud Enough To Knock You Down.

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    • #92
      Re: Monty Hall, we have a PROBLEM

      Twitch Boy's post reminded me of a different computer programming tournament.

      The challenge: you and another player have an opportunity to enter into a transaction. You can either decline the transaction, or agree to proceed. If you agree to proceed, do you uphold your end of the bargain, or do you take the other player's money / goods and run?

      The tournament tested a wide variety of strategies in "competition" with each other.

      Obviously, in the short run, it would appear that you come out ahead by betraying the other.

      However, in the long run, I'm sure many people could guess what the optimal strategy was, no matter how many variations and ideas were tested:

      The optimal long-term strategy was called "tit for tat." It was very simple. You always cooperate, unless you are betrayed. Once you are betrayed, you never enter into another transaction again with the player that betrayed you.

      If you do a search for the name of the winning strategy in quotation marks, you will find a number of articles describing the tests and outcomes in more detail.
      Last edited by FreshFish; 08-31-2013, 07:45 AM.
      "Hope is a good thing; maybe the best of things."

      "Beer is a sign that God loves us and wants us to be happy." -- Benjamin Franklin

      "Being Irish, he had an abiding sense of tragedy, which sustained him through temporary periods of joy." -- W. B. Yeats

      "People generally are most impatient with those flaws in others about which they are most ashamed of in themselves." - folk wisdom

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      • #93
        Re: Monty Hall, we have a PROBLEM

        There are five pirates. (We'll call them A, B, C, D, and E in order of seniority.) They come across 100 gold coins and have to figure out how to divvy them up.

        They decide to do this: Pirate A will propose a distribution. All five pirates (including A) then vote whether to accept the distribution, or throw Pirate A overboard. If they throw him overboard, Pirate B gets to propose a distribution, and the surviving pirates vote to accept it or throw him overboard, and so on. Majority rules, tie vote equals acceptance.

        The pirates are logical beings and do not trust each other (no promises, retaliation considerations, pinky swears, etc.) Their motivations in order of priority:

        1) Survive.
        2) Get as many gold coins as you can out of the deal.
        3) All other things being equal, throw someone overboard (because they're pirates and it's fun!)

        How many gold coins can A get away with proposing for himself?

        Believe it or not, A can get away with 98 coins.

        We'll work backwards. Start with just D and E. D proposes D100/E0 and there's not a darn thing E can do about it. D accepts, E rejects, proposal passes.

        Now bring back C. We know that if C gets thrown overboard, D will offer D100/E0 and E gets hosed, so C can simply offer E 1 coin to get his vote. C proposes C99/D0/E1, C and E accept, D rejects, proposal passes.

        Now bring back B. We know that if B gets tossed C will offer C99/D0/E1 and D gets hosed. B can offer D 1 coin to obtain his vote, and doesn't have to worry about C or E (who have motivation to toss him anyway.) B proposes B99/C0/D1/E0, B and D accept, C and E reject, proposal passes.

        Now bring back A. We know that if A gets tossed B will offer B99/C0/D1/E0, and C and E get hosed. All A has to do is offer C and E one coin each and he has their vote, and doesn't have to worry about B or D. A proposes A98/B0/C1/D0/E1, A, C, and E accept, B and D reject, proposal passes.

        You can even extend this out to an infinitely large number of pirates. Simply offer every alternating pirate 0, 1, 0, 1, etc. and yourself the rest.
        Last edited by Twitch Boy; 09-03-2013, 01:33 PM. Reason: made wording clearer
        Michigan Tech Huskies Pep Band: There's No Use Trying To Talk. No Human Sound Can Stand Up To This. Loud Enough To Knock You Down.

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        • #94
          Re: Monty Hall, we have a PROBLEM

          Here is a famous contemporary unsolved puzzle: the Kryptos sculpture outside CIA headquarters.

          Kryptos is a sculpture located on the grounds of CIA Headquarters in Langley, Virginia. Installed in 1990, its thousands of characters contain encrypted messages, of which three have been solved (so far). There is still a fourth section at the bottom consisting of 97 characters which remains uncracked. This webpage contains some information about the sculpture, including some photos collected from around the web, some rubbings of the sculpture taken by your intrepid webmistress, links to other articles and Kryptos discussion groups here and there, and information about other encrypted sculptures which have been created by the sculptor, Jim Sanborn.
          http://www.elonka.com/kryptos/
          "Hope is a good thing; maybe the best of things."

          "Beer is a sign that God loves us and wants us to be happy." -- Benjamin Franklin

          "Being Irish, he had an abiding sense of tragedy, which sustained him through temporary periods of joy." -- W. B. Yeats

          "People generally are most impatient with those flaws in others about which they are most ashamed of in themselves." - folk wisdom

          Comment


          • #95
            Re: Monty Hall, we have a PROBLEM

            Originally posted by Twitch Boy View Post
            There are five pirates. (We'll call them A, B, C, D, and E in order of seniority.) They come across 100 gold coins and have to figure out how to divvy them up.

            They decide to do this: Pirate A will propose a distribution. All five pirates (including A) then vote whether to accept the distribution, or throw Pirate A overboard and let Pirate B propose. Majority rules, tie vote equals acceptance.

            The pirates are logical beings and do not trust each other (no promises, retaliation considerations, pinky swears, etc.) Their motivations in order of priority:

            1) Survive.
            2) Get as many gold coins as you can out of the deal.
            3) All other things being equal, throw someone overboard (because they're pirates and it's fun!)

            How many gold coins can A get away with proposing for himself?
            I LOVE this problem. However I'm perplexed about what to do if you can't extend the "accept / throw overboard" option for Pirate B, C, .... By omitting that do you really mean it can't be extended on down to the 2nd-to-last Pirate?

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            • #96
              Re: Monty Hall, we have a PROBLEM

              Originally posted by owslachief View Post
              I LOVE this problem. However I'm perplexed about what to do if you can't extend the "accept / throw overboard" option for Pirate B, C, .... By omitting that do you really mean it can't be extended on down to the 2nd-to-last Pirate?
              Yeah, I should have been more clear about that. That's how it works. If A gets tossed, B gets to propose and the surviving pirates vote to accept or toss him. Rinse and repeat until a proposal is accepted.
              Michigan Tech Huskies Pep Band: There's No Use Trying To Talk. No Human Sound Can Stand Up To This. Loud Enough To Knock You Down.

              Comment


              • #97
                Re: Monty Hall, we have a PROBLEM

                Originally posted by Twitch Boy View Post
                Yeah, I should have been more clear about that. That's how it works. If A gets tossed, B gets to propose and the surviving pirates vote to accept or toss him. Rinse and repeat until a proposal is accepted.
                Interesting little thought experiment, but it sure doesn't translate to real life. If I'm pirate E, I'm *definitely* voting against pirate A if he proposes to keep 98 coins - I'd very happily "spend" my one coin just to bugger that selfish SOB. So if the question is posed as "how many coins can A get away with proposing for himself?" then the answer is absolutely not 98. I'd rather see D walk away with 100 when it comes down to the two of us - I couldn't blame D for taking all 100, but I absolutely have a problem with A trying to take 98 for himself first. One of your premises is that the pirates can't trust each other; pirate A certainly shouldn't trust me to take my one stinking coin if I'm pirate E even if that's the "rational" thing to do. In fact, in the real world, I would think that pirate A would need to propose something less than 20 coins for himself in order to ensure a majority vote. Even if he proposes an even split of 20 each, the remaining 4 would probably all think they can get more by splitting it fewer ways.

                Edit: Also, E has no skin in the game - he can't be tossed overboard, because he'll never have to make a proposal. So if the best you offer him is 1 coin better than his status quo when they are 100 on the table, I'd predict a "no" vote virtually every time.
                Last edited by LynahFan; 09-03-2013, 01:57 PM.
                If you don't change the world today, how can it be any better tomorrow?

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                • #98
                  Originally posted by LynahFan View Post
                  Interesting little thought experiment, but it sure doesn't translate to real life. If I'm pirate E, I'm *definitely* voting against pirate A if he proposes to keep 98 coins - I'd very happily "spend" my one coin just to bugger that selfish SOB. So if the question is posed as "how many coins can A get away with proposing for himself?" then the answer is absolutely not 98. I'd rather see D walk away with 100 when it comes down to the two of us - I couldn't blame D for taking all 100, but I absolutely have a problem with A trying to take 98 for himself first. One of your premises is that the pirates can't trust each other; pirate A certainly shouldn't trust me to take my one stinking coin if I'm pirate E even if that's the "rational" thing to do. In fact, in the real world, I would think that pirate A would need to propose something less than 20 coins for himself in order to ensure a majority vote. Even if he proposes an even split of 20 each, the remaining 4 would probably all think they can get more by splitting it fewer ways.

                  Edit: Also, E has no skin in the game - he can't be tossed overboard, because he'll never have to make a proposal. So if the best you offer him is 1 coin better than his status quo when they are 100 on the table, I'd predict a "no" vote virtually every time.
                  By pure logic, Twitch's answer is correct, except we know that pirates (or people) in that case will never act completely rational.

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                  • #99
                    Re: Monty Hall, we have a PROBLEM

                    A hot topic that seems to be making the rounds on Facebook the last couple days is "Take the High School Quiz", which tests an adult's ability to answer questions that a current high school senior be able to answer, in a variety of areas (chemistry, algebra, physics, etc.).

                    One of the questions (to which I am sure many adults would leap to an incorrect conclusion) is this:

                    Q: A horse runs a two-lap race around a circular track. During the first lap, its average speed is 20 miles per hour. What must the horse's average speed be during the second lap so its average speed over the course of the entire two-lap race is 40 m.p.h.?

                    In light of the method used to answer the above question, I'll pose another one (see below, not on the Quiz). The parameters are different, but should the thinking used to solve this one be the same?

                    Q: Larry is missing a concert ticket to a show he's been waiting months to see. If he doesn't leave his home in half an hour, he won't get there on time. The only thing he knows for a certainty is it's in a pocket in a pair of pants he owns. He also knows that he hasn't done laundry since he bought the ticket a week ago. The one thing about Larry, though, is when something goes missing, it always ends up being in the last likely place he mentally lists, which is this: He owns 11 pairs of pants, all of which have four pockets (two front, two back).

                    He now goes on his search, and for the first 15 minutes, he manages to search at an average rate of 40 pockets per hour. Is it possible for him to find his concert ticket on time?



                    Bonus: What is the ESSENTIAL difference between the first and second question?
                    Last edited by owslachief; 09-06-2013, 12:45 PM.

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                    • Re: Monty Hall, we have a PROBLEM

                      Originally posted by owslachief View Post
                      A hot topic that seems to be making the rounds on Facebook the last couple days is "Take the High School Quiz", which tests an adult's ability to answer questions that a current high school senior be able to answer, in a variety of areas (chemistry, algebra, physics, etc.).

                      One of the questions (to which I am sure many adults would leap to an incorrect conclusion) is this:

                      Q: A horse runs a two-lap race around a circular track. During the first lap, its average speed is 20 miles per hour. What must the horse's average speed be during the second lap so its average speed over the course of the entire two-lap race is 40 m.p.h.?

                      In light of the method used to answer the above question, I'll pose another one (see below, not on the Quiz). The parameters are different, but should the thinking used to solve this one be the same?

                      Q: Larry is missing a concert ticket to a show he's been waiting months to see. If he doesn't leave his home in half an hour, he won't get there on time. The only thing he knows for a certainty is it's in a pocket in a pair of pants he owns. He also knows that he hasn't done laundry since he bought the ticket a week ago. The one thing about Larry, though, is when something goes missing, it always ends up being in the last likely place he mentally lists, which is this: He owns 11 pairs of pants, all of which have four pockets (two front, two back).

                      He now goes on his search, and for the first 15 minutes, he manages to search at an average rate of 40 pockets per hour. Is it possible for him to find his concert ticket on time?



                      Bonus: What is the ESSENTIAL difference between the first and second question?
                      Larry needs to do laundry more often. He also shouldn't mix his dirty and clean clothes either.

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                      • Re: Monty Hall, we have a PROBLEM

                        Originally posted by owslachief View Post
                        A hot topic that seems to be making the rounds on Facebook the last couple days is "Take the High School Quiz", which tests an adult's ability to answer questions that a current high school senior be able to answer, in a variety of areas (chemistry, algebra, physics, etc.).

                        One of the questions (to which I am sure many adults would leap to an incorrect conclusion) is this:

                        Q: A horse runs a two-lap race around a circular track. During the first lap, its average speed is 20 miles per hour. What must the horse's average speed be during the second lap so its average speed over the course of the entire two-lap race is 40 m.p.h.?

                        In light of the method used to answer the above question, I'll pose another one (see below, not on the Quiz). The parameters are different, but should the thinking used to solve this one be the same?

                        Q: Larry is missing a concert ticket to a show he's been waiting months to see. If he doesn't leave his home in half an hour, he won't get there on time. The only thing he knows for a certainty is it's in a pocket in a pair of pants he owns. He also knows that he hasn't done laundry since he bought the ticket a week ago. The one thing about Larry, though, is when something goes missing, it always ends up being in the last likely place he mentally lists, which is this: He owns 11 pairs of pants, all of which have four pockets (two front, two back).

                        He now goes on his search, and for the first 15 minutes, he manages to search at an average rate of 40 pockets per hour. Is it possible for him to find his concert ticket on time?



                        Bonus: What is the ESSENTIAL difference between the first and second question?
                        Well, since you phrased it as "is it *possible*" for him to find the ticket?" - of course it is possible, since it might be in the very next pocket he searches. Unless you meant to stipulate that it will definitely be in the 44th pocket that he gets to, in which case he won't have time. I don't really understand how this is supposed to be a trick question.
                        If you don't change the world today, how can it be any better tomorrow?

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                        • Re: Monty Hall, we have a PROBLEM

                          Originally posted by LynahFan View Post
                          Well, since you phrased it as "is it *possible*" for him to find the ticket?" - of course it is possible, since it might be in the very next pocket he searches. Unless you meant to stipulate that it will definitely be in the 44th pocket that he gets to, in which case he won't have time. I don't really understand how this is supposed to be a trick question.
                          Not a trick question - all the premises are there in the question. Yes, you can conclude that the 44th pocket contains the ticket. Also, note the phrase "for the first 15 minutes".
                          Last edited by owslachief; 09-06-2013, 01:47 PM.

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                          • Re: Monty Hall, we have a PROBLEM

                            Originally posted by owslachief View Post
                            A hot topic that seems to be making the rounds on Facebook the last couple days is "Take the High School Quiz", which tests an adult's ability to answer questions that a current high school senior be able to answer, in a variety of areas (chemistry, algebra, physics, etc.).

                            One of the questions (to which I am sure many adults would leap to an incorrect conclusion) is this:

                            Q: A horse runs a two-lap race around a circular track. During the first lap, its average speed is 20 miles per hour. What must the horse's average speed be during the second lap so its average speed over the course of the entire two-lap race is 40 m.p.h.?

                            In light of the method used to answer the above question, I'll pose another one (see below, not on the Quiz). The parameters are different, but should the thinking used to solve this one be the same?

                            Q: Larry is missing a concert ticket to a show he's been waiting months to see. If he doesn't leave his home in half an hour, he won't get there on time. The only thing he knows for a certainty is it's in a pocket in a pair of pants he owns. He also knows that he hasn't done laundry since he bought the ticket a week ago. The one thing about Larry, though, is when something goes missing, it always ends up being in the last likely place he mentally lists, which is this: He owns 11 pairs of pants, all of which have four pockets (two front, two back).

                            He now goes on his search, and for the first 15 minutes, he manages to search at an average rate of 40 pockets per hour. Is it possible for him to find his concert ticket on time?



                            Bonus: What is the ESSENTIAL difference between the first and second question?
                            My answers are in white:

                            The first one is what we used to call an "old chestnut"; I've seen it in many forms over the years. The horse has to complete the second lap instantaneously.

                            The second question, if we assume that the ticket is for sure in one of the 11 pairs of pants, means that he MIGHT be able to make it to the concert on time, as long as his travel time to the concert is longer than thirty-six minutes and he isn't the driver, because he can put on a pair of pants without searching the pockets in that pair until after they are already on their way, and he also can carry five pairs of unsearched pants with him and search those on his way to the concert.
                            Last edited by FreshFish; 09-06-2013, 01:54 PM.
                            "Hope is a good thing; maybe the best of things."

                            "Beer is a sign that God loves us and wants us to be happy." -- Benjamin Franklin

                            "Being Irish, he had an abiding sense of tragedy, which sustained him through temporary periods of joy." -- W. B. Yeats

                            "People generally are most impatient with those flaws in others about which they are most ashamed of in themselves." - folk wisdom

                            Comment


                            • Re: Monty Hall, we have a PROBLEM

                              Originally posted by owslachief View Post
                              A hot topic that seems to be making the rounds on Facebook the last couple days is "Take the High School Quiz", which tests an adult's ability to answer questions that a current high school senior be able to answer, in a variety of areas (chemistry, algebra, physics, etc.).

                              One of the questions (to which I am sure many adults would leap to an incorrect conclusion) is this:

                              Q: A horse runs a two-lap race around a circular track. During the first lap, its average speed is 20 miles per hour. What must the horse's average speed be during the second lap so its average speed over the course of the entire two-lap race is 40 m.p.h.?
                              Pretty sure it's impossible. Can't double your speed, if you've burnt half your distance.

                              Assume a mile track. The first lap is 20mph, so it takes three minutes.

                              If the goal is a 40 mph average, then with a two mile distance you need to average 1.5 minutes per lap. Since you've used three minutes total already, it's impossible.

                              Comment


                              • Originally posted by owslachief View Post
                                Not a trick question - all the premises are there in the question. Yes, you can conclude that the 44th pocket contains the ticket. Also, note the phrase "for the first 15 minutes".
                                It's kind of vaguely worded. Of course it's theoretically possible. I'm too lazy to do the math, but he's searched 10 pockets in 15 minutes, so he'll need to search 34 over the next 15 minutes. Logically possible as a purely mathematical question? Sure. Realistically possible? Not enough information available to make that call.

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