in reply to Boolean math: Fill in the blanks.

I think that a solution for 21 is ( R | R ) & ( R | R | R ), but the method I used is not directly extensible to the other two cases.

Let's say that En is an expression formed or'ing and and'ing R's together, and let's call P(En) the probability that a bit is 1. So P(R)=1/2; P(R|R)=3/4 and so on. So we find that:

So if you want P(Ex)=21/32 it's easy, because 21/32=7/8*3/4 and I know (from your list) that P(R|R|R)=28/32=7/8 P(R|R)=24/32=3/4.

But it's not applicable to 23/32 (23 being prime) nor to 27/32 for you can'e decompose this fraction in the product of integer fractions of value<1.

Rule One: "Do not act incautiously when confronting a little bald wrinkly smiling man."

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Re^2: Boolean math: Fill in the blanks.
by pjotrik (Friar) on Oct 10, 2008 at 10:47 UTC
    I totally agree with you on these formulas.

    Using the second one, I get 23/32 = 28/32 - 5/32 = 1/4 + 5/8 - 5/32, therefore 23 => R & R | (R & R | R).

    I'll try this with 27 as well...
    UPDATE: 27/32 = 3/4 + 3/8 - 9/32, therefore 27 => R | R | (R | R) & R

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Re^2: Boolean math: Fill in the blanks.
by BrowserUk (Patriarch) on Oct 10, 2008 at 10:23 UTC
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      Solution for 27 would be easy: 27=32-5; 5*2=10; 22=32-10; so P(Ex|R)=27/32 if P(Ex)=22.

      But we don't have a solution for 22, because yours is wrong :)

      Update: Got it. 22=(R|R)&R|R; 27=(R|R)&R|R|R

      Rule One: "Do not act incautiously when confronting a little bald wrinkly smiling man."

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        Nice++ Thankyou, I now have the full set.

        Except...it can be taken further: R & R & R & R & R & R gives 0.5 bits :)

        So, it should be possible to get all 32 n.5 possibilities. And then all n.(25|75) etc.

        The ultimate question is this. Given an arbitrary target in the range 0 .. 99.99999999%..., is it possible to programmically generate the sequence required to approximate it?

        Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
        "Science is about questioning the status quo. Questioning authority".
        In the absence of evidence, opinion is indistinguishable from prejudice.
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        But we don't have a solution for 22, because yours is wrong :)

        Indeed. OP annotated to that effect.

        In theory, it should be possible to work backward from the solution for 11:R & R & R | R & R by removing an & R term...but it appears to be not so easy.

        Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
        "Science is about questioning the status quo. Questioning authority".
        In the absence of evidence, opinion is indistinguishable from prejudice.
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