Probability of 4 balls in trash can

[Dell]

4 balls are placed randomly into 10 bins,
what is the probability that exactly 8 of the bins will be empty??

first i need to find out how many ways i can place the balls into the bins --> 10*10*10*10=10000

now i need to find how many ways the balls can be placed so that there are 8 empty bins- or in other words that there are 2 not empty bins

so i choose 2 random bins out of the 10
10C2 = 45

now i need to place the 4 balls
1, 3
2, 2
3, 1

all in all 3 ways

45*3/10000=0.0135,,

the correct answer is 0.63

 

[Dell]

sorry 0.063, but still not my answer

 

[ystael]

Your choice of whether the balls are distinguishable or indistinguishable is inconsistent. You compute the total number of possibilities $$10^4$$ as if the balls are distinguishable, so that the sample space consists of a choice of bin for each ball, and a sample point can be represented by an ordered quadruple of integers between 1 and 10. However, when you get to describing possible arrangements for the balls, you count only three possible distributions, taking into account only the number of balls in each chosen bin, and not their identity; this assumes the balls are indistinguishable. You can get the right answer for the probability with either choice (distinguishable or indistinguishable balls), but you must be consistent and use the same choice to compute the size of the sample space and the number of successful sample points.

You also make a mistake by considering the distributions (1, 3), (2, 2), and (3, 1) to be equiprobable; they are not, as you can see most clearly by considering distinguishable balls.

 

[Dell]

to tell you the truth, I am not sure what you mean,

i think that the balls are indistinguishable, but i still have 10 options to place each one don't i? otherwise how would i find the sample space?

i am not very good at this, and this question is driving me crazy.

 

[ystael]

If the balls are indistinguishable, then there are fewer than $$10^4$$ points in the sample space, because "put the first three balls in bin 1, the fourth in bin 2" and "put the first ball in bin 2, the last three in bin 1" are the same sample point. You should probably proceed assuming the balls are distinguishable, because that makes the counting easier, if you don't know the trick for indistinguishable balls.

 

[Dell]

okay, so then i have 4 options right?

1,3 3,1 2,2 2,2

but that still won't give me the right answer

 

[Dell]

10c2*4/10^4=0.018