If are independent random variables that are uniformly distributed over compute the probability that the largest of the three is greater than the sum of the other two.
The probability density for 1 variable is the function:
p(x) = 1 if 0<=x<=1 and 0 otherwise.
The probability density for the variable t = x+y = sum of 2 variables having the above probability density is the function:
p(t) =
t if 0<=t<=1 and
2-t if 1<=t<=2.
Since the 3rd variable (call it w) has a max value of 1, the part of the probability distribution for t going as 2-t is irrelevant - the 3rd variable w has probability 0 that it will be greater than t.
For the other part of the distribution, 0<=t<=1, the probability that w > t is 1-t. That means we can calculate the probability that w > t =