Question

A die is loaded in such a way that the probability of the face with j dots turning up is proportional to j for j=1,2,3,4,5,6.in 6 independent throws of this die,what is the probability that each face turns up exactly once?

Answer 1

let the probability of face with 1 dot turning up be 1k
the probability of face with 2 dots turning up 2k
the probability of face with 3 dots turning up 3k 
and so on 
but since an outcome should be any one the face the sum of all these should be 1

so,1k+2k+3k+4k+5k+6k=1 
that is 21k=1 
we get k= 1/21 
now the probability that each face turns up exactly once in 6 throws is 
1k*2k*3k*4k*5k*6k
that is  
so the answer is  or Factorial 6/21^6

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