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Waaay OT Late Night Head Challenge for Math Gurus...

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  • Waaay OT Late Night Head Challenge for Math Gurus...

    I ran across this brain teaser and was going to dismiss it post haste. However, it struck me that this is a real life scenario that the military faces all the time. I'm guessing that some similar number crunching is going on with the oil spill in trying to calculate various probabilities and possibilities there.

    I can hold my own up through Trig, but I'm still waiting for that damn train from Topeka to pass the one from Oshkosh somewhere near Mt. Rushmore. I don't even know how to begin the following as I always sucked at probability - unless the dice were loaded or the coin had 2 heads. Any thoughts?

    ***

    You are the Intelligence Officer for a Special Ops unit planning to assault a terrorist hideout and rescue several kidnapped citizens of your country. Your Commanding Officer has asked you to calculate the probability that the kidnapped citizens will be released as a result of the assault.

    Construct a probability tree considering the following:

    1. The success or failure of the Special Ops Unit reaching the terrorist hideout.

    2. An attempt by the terrorists (yes or no) to booby-trap the kidnapped citizens with explosives.

    3. The killing of the kidnapped citizens (none, some, all) during the rescue attempt.

    You have carefully analyzed the situation and determined there is a .9 probability of the Special Operations Unit reaching the terrorist hideout and a .3 probability that the Kidnapped citizens will be booby-trapped with explosives.

    If the kidnapped citizens are booby trapped, there is a .4 probability that all will be killed during the assault, a .4 probability that some will be killed, and a .2 probability that none will be killed.

    If the kidnapped citizens are not booby-trapped, there is a .6 probability that none will be killed, a .2 probability that some will be killed, and a .2 probability that all will be killed.

    Calculate the total probability that all of the kidnapped citizens will be rescued, and provide any other finding you believe relevant.

    ***

    I was fine until the part about the "probability tree" (third sentence). Is that anything like an oak or maple?

    Would anyone out there dare hazzard a guess? How would one even begin to tackle this?

    Thanks,

    Phil

  • #2
    Too long

    Didn't read
    Mike

    My Dad always said, "If you want people to do things for you on the farm, you have to buy a machine they can sit on that does most of the work."

    Comment


    • #3
      It's pretty easy actually.

      So determine the probabilities, just multiply them together.

      For example, the troops have a .9 probability of getting to the terrorist's hideout. There's a .3 chance the hostages are wired. That means there's a .27 chance the troops will arrive to find wired hostages. There's a .63 chance they'll arrive to find un-wired hostages. Then you repeat the method for the next layer of probabilities. Since there's a .27 chance of finding wired hostages and .4, .4, and .2 chance of all, some, or none of the hostages being killed, the chances of the events are .27*.4, .27*.4, and .27*.2 respectively, or .11, .11, and .05. So there's a grim 5% chance you'll get all the hostages out alive if they're wired.

      Applying this to the unwired part of the problem, the final chances are .13, .13, and .38 for all, some, or none, respectively.

      *assuming the wired and unwired situations are unrelated events*, you can sum the like results. So the chances of losing all the hostages is .11+.13 = .24. Same for some being killed .11+.13. However, getting them all out is a .05+.38 = .43 chance.

      So if you do a single assault and you know nothing about the situation you can expect all the hostages to live 43% of the time. You'll lose them all 24% of the time. You'll lose some 24% of the time. 43+24+24 = 91 (rounding error, it's really 90) The reason it's 90 instead of 100 is because 10% of the time nothing happens - the troops get lost.

      Again, this assumes the branches of the tree are independent. For example, say the troops get lost but the terrorists have spies that tell them the troops are in the area and they kill the hostages - those are no longer independent probabilities.

      Regarding the .1 probability of the troops getting lost, I don't count this one way or the other since it doesn't help resolve the situation.

      Now, if I were the commander, if the troops get lost, I'd sent them back in. They will eventually find the hideout. So the .9 probability would become 1 - a 100% chance of finding the hideout. The odds change because now we're multiplying by 1 instead of .9. In this case, all the hostages die 26% of the time, some die 26% of the time, and all live 48% of the time. Same proportions.
      Last edited by Tony Ennis; 06-03-2010, 09:41 PM.

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      • #4
        I'd like to know how they came up with the initial probabilities.

        I agree with Tony's results. I didn't calculate it out, but his method would have been the one I followed.

        andy b.
        The danger is not that computers will come to think like men - but that men will come to think like computers. - some guy on another forum not dedicated to machining

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        • #5
          Phil, very easy Q if you draw it out as a decision tree...excel is excellent for this. Its more easy to get muddled without the tree. The become a usefull business tool, er sales tool, when there's differing dollar amounts at each decision point. Of course the probabilities are pulled from thin air but most won't argue with something as serious sounding as biz science....hence its main function is as a sales tool to get what you want

          like this: http://www.treeplan.com/images/treep...6_13-22-46.gif
          in Toronto Ontario - where are you?

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          • #6
            I'd have just cracked open a Bud and waited for the 11:00 o'clock news. What's the rush?

            Comment


            • #7
              Originally posted by andy_b
              I'd like to know how they came up with the initial probabilities.

              I agree with Tony's results. I didn't calculate it out, but his method would have been the one I followed.

              andy b.
              In my opinion that is the problem with a lot of decision making. You can do wonderful things with mathematics but if the initial premise is wrong or of unknown accuracy, then so is the result, no matter how brilliant your calculations are.

              I do understand this is a theoretical situation where the initial conditions are givens and are therefore assumed correct.

              Don Young
              Last edited by Don Young; 06-03-2010, 11:10 PM.
              Don Young

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              • #8
                I wouldn't give a nickle's worth of thought to any of their outcomes.

                Life and death situations are not 24 Hours things.

                "The Best Laid Plans of Mice And Men."

                Thing do NOT go according to plan. That is why we are at Plan H or something with the Gulf Spill. Seems we will be getting to AA or BB or MM before the leak is fixed.

                ALL of this was supposedly planned to prevent, tho' Transocean says it was denied by BP.

                Cheers,

                George

                Comment


                • #9
                  Even when the probabilities are made up there's a value to it. Once they are on paper, they can be discussed and debated. They'll also tell which parts of the decision tree are more important than others so resources can be spent to determine more realistic estimates.

                  Comment


                  • #10
                    Probability is a field where most people get lost immediately. It is not intuitive at all. If you flip a coin 10 time and it comes up heads ten times in a row then what is the chance of it coming up heads on the next flip?


                    It is still 50/50. The coin has no memory. That is a point that if very often misunderstood.

                    Another example is the Monty Hall Problem.

                    Imagine that the set of Monty Hall's game show Let's Make a Deal has three closed doors. Behind one of these doors is a car; behind the other two are goats. The contestant does not know where the car is, but Monty Hall does.

                    The contestant picks a door and Monty opens one of the remaining doors, one he knows doesn't hide the car. If the contestant has already chosen the correct door, Monty is equally likely to open either of the two remaining doors.

                    After Monty has shown a goat behind the door that he opens, the contestant is always given the option to switch doors. What is the probability of winning the car if she stays with her first choice? What if she decides to switch?
                    Free software for calculating bolt circles and similar: Click Here

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                    • #11
                      Originally posted by Evan
                      Probability is a field where most people get lost immediately. It is not intuitive at all. If you flip a coin 10 time and it comes up heads ten times in a row then what is the chance of it coming up heads on the next flip?


                      It is still 50/50. The coin has no memory. That is a point that if very often misunderstood.
                      A true probabilities wank would suspect a forcing of the results and bet the trend. Or at least he'd convince his customers to bet that way.

                      Comment


                      • #13
                        Tiffie,

                        Wikipedia is filled with trash. You cannot depend on what is posted there. It serves no purpose to post a bunch of links to what anybody on this site can look up if they are so inclined. If you want explanations for math questions try Wolfram Mathworld.

                        http://mathworld.wolfram.com/
                        Free software for calculating bolt circles and similar: Click Here

                        Comment


                        • #14
                          Themes

                          Evan,

                          unless you can prove otherwise and conclusively that Wikipedia is full of trash (to the exclusion of all other as you say and infer) is pure subjectivity - which you, like anyone else, are quite entitled to have as they are to judge for themselves.

                          There was a lot of unsubstantiated and non-defined terminology being tossed around, so I decided to post a few links that I think are relevant and that others are able and free to assess for themselves and (to) use and add to - or not - as it suits them.

                          You, as are any others who choose to do so, are quite entitled to edit or seek to have Wikipedia items edited and revised and judged by you peers.

                          I look forward to the result of your contributions to Wikipedia.

                          I have used Wolfram before, and went to the link you posted and (then) to Wolframs Probability and Statistics links at:
                          http://mathworld.wolfram.com/topics/...tatistics.html

                          Perhaps, as it seems that you have endorsed Wolfram in that context, that you can show us - or me - where and in what ways Wolfram over-rides and contradicts the Wikipedia links that I posted.

                          I await your learned discourse which will - or should be able to - prove conclusively that Wikipedia is "filled with trash" - including the topics of probability and statistics which are the theme/s of this thread.

                          Comment


                          • #15
                            Im still laughing at Tiffers title, I tuned in this morning and in the general category it read "Waaay OT Late Night Head..."

                            I said to myself - wow --- this really is going to be Waaaaaaaay OT.

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