How Many Ways Can You Roll Six Dice with Repeating Numbers?

[ArcanaNoir]

Shouldn't the probability that all numbers are different plus exaclt 2 are alike and the rest different plus exactly three are alike plus...up to they are all the same be no greater than one?

 

[I like Serena]

Shouldn't the probability that all numbers are different plus exaclt 2 are alike and the rest different plus exactly three are alike plus...up to they are all the same be no greater than one?

Correct, they should sum up to exactly one (or less than one depending on what you calculate exactly).

 

[ArcanaNoir]

Okay, I think I have this problem all done. I've solved it probably more than one way by now, and I got consistent answers. *runs away*

 

[Ray Vickson]

Homework Statement

How many ways can you roll six dice so that at least 2 numbers are the same? At least 3? At least 4? At least 5?

Homework Equations

The Attempt at a Solution

I've used every equation in the chapter and filled page after page with numbers all moved around. I'm drowning! Please help. I hate probability.

{at least two the same} = E2 + E3 + E4 + E5 + E6, where Ej = {exactly j the same} (and "+" denotes set union). E2 = {2 1s, others all different}+{2 2s, others all different}+... Clearly, all these have the same number of elements, so the number of elements in E2 is 6*|{2 1s, all others different}| (|{.}| = set cardinality). Now think of 6 bins and 6 balls tossed at random into them; we want the number of outcomes in which bin 1 has 2 balls and the others all have 0 or 1 ball each. This the type of thing for which the _multinomial_ distribution was designed.

RGV

 

[I like Serena]

*runs away*

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