One of the numbers 1, 2, 3, … , 100 is picked at random. What is the probability that it is either a perfect square or a perfect cube?

There are 10 perfect squares (1, 4, 9, … , 100) and 4 perfect cubes (1, 8, 27, 64). Two numbers sit on both lists — 1 and 64, since $64=8^2=4^3$ — so the union holds 10 + 4 − 2 = 12 numbers and the probability is 12/100 = 3/25. Adding the counts with no subtraction gives 14/100 = 7/50; removing only the obvious repeat 1 and overlooking 64 gives 13/100; and 1/10 counts the squares alone.