Need a math pro to answer two questions

[pcity]

If you're asking about the case where each color die rolls the same number twice in a row, 1/216 is correct for the first one.

Assume red die, green die, blue die. Assume that on first roll, red die rolls x, green die rolls y, blue die rolls z.

Chances of getting x on the second roll of red die = 1/6.

Chances of getting y on the second roll of green die = 1/6

Chances of getting z on the second roll of blue die = 1/6.

They're each independent events so you can just multiply the probabilities, (1/6)* (1/6)*(1/6) = 1/216

[Ruffin_Ready]

Originally Posted by suzy_q2010

Question 1:

The two rolls are independent events. Rolling the second time is not influenced by the first.

We always assume the dice are fair.

Roll once, you have the first 3 numbers.

For 3 dice, there are 6 times 6 times 6 possible combinations of numbers = 216 .

The odds of the second roll matching the first are 1 in 216.

Color does not make a difference.

Suppose the first roll is red die = 1, white die = 2, and blue die = 3.

On the second roll, the odds the red die will be a 1 are still 1 in 6, the white die will be a 2 is 1 in 6, and the blue die a 3 is still 1 in 6. 1/6 times 1/6 times 1/6 is still 1/216.

I think you misunderstood the OP.

He said his dice were different colors. No mention of them being numbered. So I am assuming they are merely blank cubes of three differing shades.

Ergo, we need not multiply by six for each die, but rather only by one, as in one color.

So....3 dieX3 colorX3combo options=27=26-1 odds of replication of 1st throw on 2nd throw.

And as you said, subsequent tosses have the same odds and are not influenced, nor are the odds any longer than the previous throw, even if it has happened 100 times.

Regarding Q #2-- we are dealing with the age-old "theory of infinitesimals" which was perhaps most famously used in Zeno's Paradox.
Thus: yes, hypothetically, your uber-polyglon could have an infinite amount of sides--but I think the term "facets" might be a bit more geometrically accurate. This actually is not too far removed from some of the various quantum mechanics models which speak of atoms not really consisting of the old "solar system" model (Rutherford's standard model) but rather of a nebulous electron cloud hovering in at-times multiple states around the nucleus. Exact position of the electron shells is not determined until observance takes place. (see: Schrodinger's Cat).