On any day the probability of rain in a town is 0.4. On a rainy day the probability that Meera reaches school late is 0.5, while on a dry day it is only 0.2. What is the probability that she reaches school late on a randomly chosen day?

Split the day by the weather: P(late) = P(rain) × P(late given rain) + P(dry) × P(late given dry) = 0.4(0.5) + 0.6(0.2) = 0.20 + 0.12 = 0.32. Answering 0.2 keeps only the rainy branch and ignores that she can be late on a dry day too; 0.35 is the plain average of 0.5 and 0.2, which would be right only if rainy and dry days were equally likely; and 0.7 simply adds the two conditional probabilities, a sum that can even exceed 1 and so cannot be a probability at all.