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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?
Solution
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`.
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