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RE: [hr100] Odds?

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  • Crone, Brian Keith
    The problem is that when you have a draw where people can have more than one ticket the math is not so simple. Suppose there are 1000 tickets in the pot and
    Message 1 of 31 , Dec 13, 2012
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      The problem is that when you have a draw where people can have more than one ticket the math is not so simple.

      Suppose there are 1000 tickets in the pot and that people have either 1,2,4,8,16,32,or 64 tickets. I am considering the never been in pool where the number of tickets is 2^n, and therefore you get 1,2,4,8,16,32,... tickets for n=0,1,2,3,4,5,...

      If you have 4 tickets your chance of being selected in the first round is 4/1000.
      If you are not selected your chance of being selected in the second round is not necessarily 4/999.
      It is 4/(1000-m), where m is the number of tickets of the person selected in the first round, but the problem is that you don't know in advance what m is, it can be 1,2,4,8,16,32,or 64. To get the chances of getting in you have to consider all the possibilities. The same applies for each round. It becomes a mess quickly.

      I believe Blake has approximated this in the past with monte carlo simulations. You could also just work through all the options, but regardless the simple methods presented here will provide an underestimate of the odds of getting in since they assume only one ticket is taken out per round, but it will in general be more.

      Best,

      Brian

      -----Original Message-----
      From: hr100@yahoogroups.com [mailto:hr100@yahoogroups.com] On Behalf Of Jeff List
      Sent: Thursday, December 13, 2012 7:45 AM
      To: hr100@yahoogroups.com
      Subject: Re: [hr100] Odds?




      Joe's odds are a little higher than I get at the higher ticket numbers.
      Below is what I get for all three lottery pools.
      I followed more or less the same methodology as Chris Martin describes, except after each winner's draw I removed the average number of tickets/person, not just one ticket (because once a person wins, the rest of their tickets are essentially removed too).
      No guarantee these are right -- I'm definitely no expert in probability.

      Vets:
      #tickets chances(%)
      5 62.1
      6 68.9
      7 74.5
      8 79.2
      9 83.0
      10 86.1
      11 88.7
      12 90.8
      13 92.5
      14 93.9
      16 96.0
      17 96.8
      18 97.4

      Never:
      #tickets chances(%)
      1 2.0
      2 3.9
      3 5.8
      4 7.7
      5 9.5
      8 14.8
      9 16.5
      10 18.2
      16 27.5
      32 47.6
      33 48.6
      64 72.8

      Else:
      #tickets chances(%)
      1 12.7
      2 23.9
      3 33.6
      4 42.1
      5 49.6
      6 56.1

      >________________________________
      > From: Joe Lea <joelea@... <mailto:joelea%40gmail.com> >
      >To: bpwrlc@... <mailto:bpwrlc%40comcast.net>
      >Cc: hr100@yahoogroups.com <mailto:hr100%40yahoogroups.com>
      >Sent: Thursday, December 13, 2012 8:42 AM
      >Subject: [hr100] Odds?
      >
      >
      >
      >
      >Didn't see any discussion of our odds yet. 638 tickets in the "Everyone
      >Else" group. 70 slots... assuming drawn tickets are set aside...
      >
      >1 ticket = 11.61%
      >2 = 23.22%
      >3 = 34.84%
      >4 = 46.45%
      >5 = 58.06%
      >6 = 69.67%
      >
      >... is that right?
      >
      >joe lea
      >chapel hill, nc
      >
      >[Non-text portions of this message have been removed]
      >
      >
      >

      [Non-text portions of this message have been removed]
    • Brad
      Oh, I most definitely agree Fred! Very fun, great social atmosphere, and the work is most gratifying. I m making some assumptions here, but I find it logical
      Message 31 of 31 , Dec 27, 2012
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        Oh, I most definitely agree Fred! Very fun, great social atmosphere, and the work is most gratifying.

        I'm making some assumptions here, but I find it logical that the reason why extra tickets are given out for Hardrock-specific trail work (or aid station captaining, etc) is to incentive-ize folks to volunteer specifically for Hardrock itself. I can find almost the same gratification by volunteering for more local trail workdays or events.

        Of course I'll be going to Hardrock to volunteer regardless of what extra tickets I earn or don't earn, because of its great character, environment and the opportunity to learn more of the terrain for when I do get in, but its a lot easier to justify a 700 mile round trip to both the wallet and fiance (or by Hardrock, wife :) when the service tickets make a (more) meaningful difference.

        Michael Bur validly pointed out that any difference in the weight of service tickets across the lotteries could very well be an intentional choice on the part of the race committee to weigh factors more heavily in favor of the "the poor sap who keeps applying but keeps losing out," which makes good sense, if that is the intent. I was initially working from the assumption that the race board would want equality of value of service tickets across the lotteries, but perhaps I could very easily be wrong.

        As Mark Oveson said, "the motivation for Never Starters has now shifted almost entirely to making sure you stay qualified and apply every single year. Trail work is only marginally helpful."
        Right now, it's a lot more than marginally helpful to those who are first or second time entries in the lottery, as one service ticket increases the odds by roughly 87% & 40%, respectively, since the odds for those folks is so much smaller to begin with. But those who are in their third year or beyond of applying get, at most, a 17% bump in their chances. Changing from [(2^N) + S] to [2^(N+S)] seems to make the bump from service tickets more equal across the Never Starter lottery, but is that a desired goal?

        I guess the main question is: Does the reason that the service ticket incentive is offered imply a goal of having that incentive be of equal value to all interested parties across the board? And if so, is that equal value goal being accomplished in the current formula?

        Anyone up for a nice long run this Saturday where we can discuss this in a better format than email? :) (Seriously - I'll be going for ~35mi, starting at 6:30a at the Apex trailhead in Golden, CO, but the route has options to cut it to ~14 or ~25.)

        God Bless,
        Brad



        --- In hr100@yahoogroups.com, Fred Ecks <fredx@...> wrote:
        >
        > Silly me; I do the trail work each year simply because it's a helluva
        > fun day! The trail work days that my local trail running group does
        > with our county are safe and sane, short, and slooooow. The Hardrock
        > trail work is really fun, involves big dangerous tools, and we get an
        > amazing amount done in a single day. Who cares about the lottery?! :-)
        >
        > - Fred
        >
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