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Question

A die is weighted such that the probability of rolling the face numbered n is proportional to n2(n=1,2,3,4,5,6). The die is rolled twice, yielding the number a and b. The probability that a<b is P such that P=12⎜ ⎜ ⎜ ⎜ ⎜1xn=1n4y2⎟ ⎟ ⎟ ⎟ ⎟, then the value of x+y is
(where [.] represents the greatest integer function)

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Solution

P(n)n2P(n)=λn2 where 6n=1P(n)=1λ(6n=1n2)=1λ[6×7×136]=1λ=191
We know that P(a<b) and P(a>b) are equally likely cases. So, P(a<b)=P(a>b)(i)
And the sum of all mutually exclusive events, P(a<b)+P(a=b)+P(a>b)=1
P(a<b)=12[1P(a=b)] (From(i))
P=12(114+24+...+6491×91)
x=6,y=91
x+y=97

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