Probability of choosing a sum from three integers
Probability
Level
3
Suppose we randomly choose three positive integers from 1 to 10 with replacement.
What is the probability that the first integer chosen is the sum of the other two integers?
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1 solution
Suppose the integer we choose has a value of p, from numbers 1 to n, Then the various combinations we need for the other two integers are 1, p-1, 2, p-2.. 3, p-3 and so on till p-1 and 1 This gives us p-1 possibilities.
There are therefore 1 + 2 + .... n-1 = n(n-1)/2 ways of choosing the first integer. After this there are 1/n^3 ways of choosing three numbers from integers 1 to n.
So the exact probability is n(n-1)/2n^3 = (n-1)/2n^2
If we set n = 10, we get the probability as 9/200