A jar starts with 5 red and 3 green beads. A bead is taken out at random, and it is then dropped back in along with 2 more beads of its own colour. A second bead is now taken out. What is the probability that this second bead is red?

The jar is not the same at the second draw, so split the work into two branches. First bead red, probability 5/8: the jar now holds 7 red among 10 beads, so a red follows with probability 7/10. First bead green, probability 3/8: the jar holds 5 red among 10, so a red follows with probability 1/2. Altogether (5/8)(7/10) + (3/8)(1/2) = 35/80 + 15/80 = 50/80 = 5/8 — exactly the probability of a red on the FIRST draw, which is no accident: topping up with whichever colour appeared leaves the chances in balance, however many times it is repeated. The options 7/10 and 1/2 are the two branches taken on their own, as though the first colour were already known, and 3/5 is the plain average of those two branches, which wrongly treats them as equally likely.