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.

Answer

`(4)/(7)`

Explanation

Since the probabilities of the faces are proportional to the numbers on them, we can take the probailities of faces 1, 2, …, 6 as k, 2k, …, 6k, respectively. Since one of the faces must occur, we have k + 2k + 3k + 4k + 5k + 6k = 1 or k = `1//21`
Therefore, the probability of occurence of an even number is 2k + 4k + 6k = 12k = 12/21 = 4/7.