If 10 quarters are distributed at random to 4 people, what is the probability that everyone receives at least 50 cents? The answer is 14,175/4^8 but I need help figuring out how to get to that answer.

Question

If 10 quarters are distributed at random to 4 people, what is the probability that everyone receives at least  50 cents? The answer is 14,175/4^8 but I need help figuring out how to get to that answer.

anonymous · Answer

Sometimes such a problem is attacked by finding the probability that one received only one quarter or none, then subtracting that probability from 1.  I don't see how to do it right now, tho.

tkhunny · Answer

Ponder this diagramme:

**|**|**|****  (2 2 2 4)

How many different arrangements of those three bars are there?

|||********** (0 0 0 10)
*|***|****|** (1 3 4 2)
*|*|*|******* (1 1 1 7)

I'm going to guess it's quite a bit fewer than you expect.  There just aren't that many degrees of freedom.

How many of those have at least 2 in each bucket?  I'm thinking 21.6% is WAY too high!  More like 3.5%, I would think.