Double or Halve Riddle












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$begingroup$


You start off with a $100 pool. Every second, your pool doubles in value with probability $p$ or gets cut in half with probability $1-p.$ If you drop below 98.76543210¢ at any point, you lose and the host of the game show laughs at you.



Furthermore, you stubbornly refuse to stop playing. What's the probability that you eventually lose? Note: Assume that you and the host are both immortal, and that thermodynamic restrictions do not prevent gameplay in the longue duree.










share|improve this question











$endgroup$

















    2












    $begingroup$


    You start off with a $100 pool. Every second, your pool doubles in value with probability $p$ or gets cut in half with probability $1-p.$ If you drop below 98.76543210¢ at any point, you lose and the host of the game show laughs at you.



    Furthermore, you stubbornly refuse to stop playing. What's the probability that you eventually lose? Note: Assume that you and the host are both immortal, and that thermodynamic restrictions do not prevent gameplay in the longue duree.










    share|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      You start off with a $100 pool. Every second, your pool doubles in value with probability $p$ or gets cut in half with probability $1-p.$ If you drop below 98.76543210¢ at any point, you lose and the host of the game show laughs at you.



      Furthermore, you stubbornly refuse to stop playing. What's the probability that you eventually lose? Note: Assume that you and the host are both immortal, and that thermodynamic restrictions do not prevent gameplay in the longue duree.










      share|improve this question











      $endgroup$




      You start off with a $100 pool. Every second, your pool doubles in value with probability $p$ or gets cut in half with probability $1-p.$ If you drop below 98.76543210¢ at any point, you lose and the host of the game show laughs at you.



      Furthermore, you stubbornly refuse to stop playing. What's the probability that you eventually lose? Note: Assume that you and the host are both immortal, and that thermodynamic restrictions do not prevent gameplay in the longue duree.







      sequence






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      share|improve this question













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          $begingroup$

          Meta: This is in effect a standard question in the theory of random walks. Does that make it off-topic as a textbook problem? I don't think so. The relevant meta-question is this one whose excellent and highest-voted answer gives this closely related question as an example of something that's a "math puzzle" rather than a "math problem".



          OK then. Assuming that there's no weird business with rounding of fractional cents, this question is equivalent to the following: You start with a number $x=7$. You repeatedly add 1 (with probability $p$) or subtract 1 (with probability $1-p$). What's the probability that you eventually reach $x=0$? (Why 7? Because 6 halvings of $100 take you to $1.5625 (above the threshold in the question) and 7 halvings take you to $0.78125 (below the threshold).)



          And the answer, which you can find in any suitable textbook, is




          The probability is 1 if $pleqfrac12$; otherwise it's $bigl(frac{1-p}{p}bigr)^7$.




          If you don't want to go out and find a suitable textbook, you can also deduce it from the accepted answer to that question I linked above: see that formula for $c_1,c_2$ and the way in which the answer to the original question is derived from it, and then put $a=-7$ instead of $a=-1$.



          Meta again: Does that mean that this should have been closed as a duplicate? I think it's just different enough not to be: you can figure out the answer to this one from the answer to that one, but you can't just see it there.






          share|improve this answer









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            1 Answer
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            active

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            $begingroup$

            Meta: This is in effect a standard question in the theory of random walks. Does that make it off-topic as a textbook problem? I don't think so. The relevant meta-question is this one whose excellent and highest-voted answer gives this closely related question as an example of something that's a "math puzzle" rather than a "math problem".



            OK then. Assuming that there's no weird business with rounding of fractional cents, this question is equivalent to the following: You start with a number $x=7$. You repeatedly add 1 (with probability $p$) or subtract 1 (with probability $1-p$). What's the probability that you eventually reach $x=0$? (Why 7? Because 6 halvings of $100 take you to $1.5625 (above the threshold in the question) and 7 halvings take you to $0.78125 (below the threshold).)



            And the answer, which you can find in any suitable textbook, is




            The probability is 1 if $pleqfrac12$; otherwise it's $bigl(frac{1-p}{p}bigr)^7$.




            If you don't want to go out and find a suitable textbook, you can also deduce it from the accepted answer to that question I linked above: see that formula for $c_1,c_2$ and the way in which the answer to the original question is derived from it, and then put $a=-7$ instead of $a=-1$.



            Meta again: Does that mean that this should have been closed as a duplicate? I think it's just different enough not to be: you can figure out the answer to this one from the answer to that one, but you can't just see it there.






            share|improve this answer









            $endgroup$


















              2












              $begingroup$

              Meta: This is in effect a standard question in the theory of random walks. Does that make it off-topic as a textbook problem? I don't think so. The relevant meta-question is this one whose excellent and highest-voted answer gives this closely related question as an example of something that's a "math puzzle" rather than a "math problem".



              OK then. Assuming that there's no weird business with rounding of fractional cents, this question is equivalent to the following: You start with a number $x=7$. You repeatedly add 1 (with probability $p$) or subtract 1 (with probability $1-p$). What's the probability that you eventually reach $x=0$? (Why 7? Because 6 halvings of $100 take you to $1.5625 (above the threshold in the question) and 7 halvings take you to $0.78125 (below the threshold).)



              And the answer, which you can find in any suitable textbook, is




              The probability is 1 if $pleqfrac12$; otherwise it's $bigl(frac{1-p}{p}bigr)^7$.




              If you don't want to go out and find a suitable textbook, you can also deduce it from the accepted answer to that question I linked above: see that formula for $c_1,c_2$ and the way in which the answer to the original question is derived from it, and then put $a=-7$ instead of $a=-1$.



              Meta again: Does that mean that this should have been closed as a duplicate? I think it's just different enough not to be: you can figure out the answer to this one from the answer to that one, but you can't just see it there.






              share|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Meta: This is in effect a standard question in the theory of random walks. Does that make it off-topic as a textbook problem? I don't think so. The relevant meta-question is this one whose excellent and highest-voted answer gives this closely related question as an example of something that's a "math puzzle" rather than a "math problem".



                OK then. Assuming that there's no weird business with rounding of fractional cents, this question is equivalent to the following: You start with a number $x=7$. You repeatedly add 1 (with probability $p$) or subtract 1 (with probability $1-p$). What's the probability that you eventually reach $x=0$? (Why 7? Because 6 halvings of $100 take you to $1.5625 (above the threshold in the question) and 7 halvings take you to $0.78125 (below the threshold).)



                And the answer, which you can find in any suitable textbook, is




                The probability is 1 if $pleqfrac12$; otherwise it's $bigl(frac{1-p}{p}bigr)^7$.




                If you don't want to go out and find a suitable textbook, you can also deduce it from the accepted answer to that question I linked above: see that formula for $c_1,c_2$ and the way in which the answer to the original question is derived from it, and then put $a=-7$ instead of $a=-1$.



                Meta again: Does that mean that this should have been closed as a duplicate? I think it's just different enough not to be: you can figure out the answer to this one from the answer to that one, but you can't just see it there.






                share|improve this answer









                $endgroup$



                Meta: This is in effect a standard question in the theory of random walks. Does that make it off-topic as a textbook problem? I don't think so. The relevant meta-question is this one whose excellent and highest-voted answer gives this closely related question as an example of something that's a "math puzzle" rather than a "math problem".



                OK then. Assuming that there's no weird business with rounding of fractional cents, this question is equivalent to the following: You start with a number $x=7$. You repeatedly add 1 (with probability $p$) or subtract 1 (with probability $1-p$). What's the probability that you eventually reach $x=0$? (Why 7? Because 6 halvings of $100 take you to $1.5625 (above the threshold in the question) and 7 halvings take you to $0.78125 (below the threshold).)



                And the answer, which you can find in any suitable textbook, is




                The probability is 1 if $pleqfrac12$; otherwise it's $bigl(frac{1-p}{p}bigr)^7$.




                If you don't want to go out and find a suitable textbook, you can also deduce it from the accepted answer to that question I linked above: see that formula for $c_1,c_2$ and the way in which the answer to the original question is derived from it, and then put $a=-7$ instead of $a=-1$.



                Meta again: Does that mean that this should have been closed as a duplicate? I think it's just different enough not to be: you can figure out the answer to this one from the answer to that one, but you can't just see it there.







                share|improve this answer












                share|improve this answer



                share|improve this answer










                answered 4 hours ago









                Gareth McCaughanGareth McCaughan

                61.7k3153238




                61.7k3153238






























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