Question

Solve the following:

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, .......6. What is the probability, in one roll of the die, that an odd number of dots will turn up?

Answer 1

Let P(j) be the probability that j dots tum up, 1 ≤ j ≤ 6

∴ P(j) = jk, where k is the constant of proportionality

∴ `sum_("j" = 1)^6 "P"("j")` = 1

∴ k + 2k + 3k + 4k + 5k + 6k = 1

∴ 21k = 1

∴ k = `1/21`

∴ the required probability = P(1) + P(3) + P(5)

= k + 3k + 5k = 9k

= `9/21   ...[because "k" = 1/21]`

= `3/7`.