A die is loaded so that the probability of a face `i` is proportional to `i ,i=1,2,... 6.` Then find the probability of an even number occurring when the die in rolled.

Explanation

Let event `A_(i)` is ''number I appears''
`therefore P(A_(i)) prop i^(2) or P(A_(i)) = lambdai^(2)`, where `lambda` is constant of proportionality.
Now, `underset(i = 1)overset(6)Sigma P(i) = 1`
`therefore underset(i = 1) overset(6)Sigma lambdai^(2) = 1`
`implies lambdaxx (6xx7xx13)/(6)=1`
`implies lambda = (1)/(91)`
`therefore P(A_(i))= (i^(2))/(91)`
`therefore` P(prime number appears) = P(2) + P(3) + P(5)
`= (2^(2))/(91) + (3^(2))/(91) + (5^(2))/(91) = (38)/(91)`