In a survey of 120 people, 60 use app P, 55 use app Q and 45 use app R; 25 use both P and Q, 20 use both Q and R, 15 use both P and R, and 10 use all three. One of the 120 people is chosen at random. What is the probability that the person uses at least two of the apps?

'At least two' means exactly two together with all three. Exactly two = (25 − 10) + (20 − 10) + (15 − 10) = 15 + 10 + 5 = 30, and 10 people use all three, so 40 people qualify and the probability is 40/120 = 1/3. Adding the three pairwise counts as they stand gives 60, i.e. 1/2, because the 10 triple users are then counted three times instead of once; 1/4 counts the 'exactly two' group only.