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Probabilities of Selecting Multiples of Fixed Numbers

3. Let N = 1000 and let S = {1, 2, ... , N}. Let D_i = {m belongs to S: i|m} for integers i between 1 and N.

a) Are the events D_2 and D_4 independent? Do the appropriate calculation to answer this question. Then explain why your answer makes sense.

b) Are the events D_4 and D_5 independent?

c) Are the events D_5 and D_6 independent?

Solution Preview

1. The events D_2 and D_4 are not independent.

P(D_2) = 1/2 is the probability that if we choose a number at random from 1 to 1000, the number will be even.
P(D_4) = 1/4 is the probability that if we choose a number at random from 1 to 1000, the number will be divisible by 4.

In order for D_2 and D_4 to be independent, we would need the following to be true:

P(D_2 n D_4) = P(D_2) P(D_4) = 1/8.

But the event D_2 n D_4 is equal to D_4 because every number divisible by 4 is even, so D_4 is a subset ...

Solution Summary

Given a set of numbers from 1 to 1000, the solution computes the probability P(n) that a randomly-selected number in this set is a multiple of n for various fixed values of n. It also determines for various fixed m and n whether P(m) and P(n) are independent.

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