Re: Another probability "paradox".

On Sep 1, 10:28 pm, Bill Taylor <w.tay...@xxxxxxxxxxxxxxxxxxxxx>
wrote:
It seems to be the season for Monty Hall, etc, again. So:

This time, you roll two balanced dice, and you're
hoping to get a "double". The probability is 1/6.
No problem so far.

Now, you roll the dice, and they roll out of your sight,
but a friend tells you "Hey, you got a six on the blue die!"
(The dice are red and blue respectively, and your friend
is very grammatical.)

The conditional probability of having got a double, given this
information, is still 1/6 , easily obtained. Still no problem;
after all, to get a double, it doesn't really matter what you
got on the blue die. He could have told you that you'd
got a one, or a two, or anything, and it wouldn't have
changed your 1/6 probability. Still no problem.

HOWEVER, next time, the dice still roll out of your sight,
but within your friend's sight, and he tells you,
"Hey, you got at least one six this time!"

Now, if you work out the conditional probabilities
based on this new information, it turns out that
the probability of having got a double has dropped
sharply, to 1/11 (!)

Why should this be so different!? How can it make
so much difference whether or not you are told
the colour of the di(c)e that got a six?

But even that isn't the real paradox! Because the thing is,
it's still the same whether he said "six" or any other number!

P( double | at least one "1") = 1/11 ;
:
:
P( double | at least one "6") = 1/11 .

That is, your friend can reduce your odds by almost 1/2
merely by telling you you got at least one of something,
which you knew you had to get anyway!!

What the *#&%! How can this be?

-- Odd Odds Bill

B l u e

1 2 3 4 5 6

1 11 12 13 14 15 16

2 21 22 23 24 25 26
R
3 31 32 33 34 35 36
e
4 41 42 43 44 45 46
d
5 51 52 53 54 55 56

6 61 62 63 64 65 66

If I know only that I have at least one 6 then I'm in row or column 6,
which means 1 double in 11 chances. But if I know that I have a blue 6
then I'm in column 6, where my chances are 1 in 6.
.

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