Primes and Probability

From: Lee Corbin (lcorbin@ricochet.net)
Date: Sat May 12 2001 - 11:38:20 MDT


Spike wrote:

>Another way to illustrate why it is best to switch doors after Monte
>opens one is by restating the game thus: Monte offers you one of the
>three doors, but before you choose, he writes a letter on a card and
>places it face down. This is the letter of a door that he knows has a
>goat behind it. He says that after you choose a door, which might be
>the one he wrote, he will reveal the door written on the card. That
>illustration may help some to understand why its best to switch, assuming
>one prize and two zonks.

I guess you know by now that you have completely changed the problem.
It wouldn't be of any advantage here to change (see Amara's original
explanation, and Eliezer was right to point out that your new version
is equivalent to the fourth problem on my list).

You also write intriguingly,

> Since we are discussing probabilities and things that can easily
> be calculated, heres one for you: suppose you choose a number
> randomly (x), you know that the probability of its being prime
> is about 1/ln(x). OK, suppose you pick two random primes and
> multiply them together to get a composite C. What is the probability
> that C+2 is prime? Is it still 1/ln(C+2) ?

> The answer to this is not trivial, and it bears directly on the
> discussion on page 87 of Damien's updated version of The Spike. spike

I'm lucky to have that book, wherein on page 87, it is related
that a certain ominous **Mr. Jones** has determined the rate at
which very large prime numbers are likely to be discovered.
Now I would think that surely the probability that C+2 is
prime is indeed proportional to 1/ln(C+2). Yes? It would be
marvelous if someone had found otherwise. What's the connection?

Lee



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