A cloth pouch P holds 3 red and 5 white beads; a second pouch Q holds 4 red and 4 white beads. One bead is taken at random from pouch P and dropped into pouch Q; a bead is then drawn at random from pouch Q. What is the probability that this bead is red?

Split on what moved across. The transferred bead is red with probability 3/8, and pouch Q then holds 5 red out of 9, giving (3/8) × (5/9) = 15/72. The transferred bead is white with probability 5/8, leaving 4 red out of 9, giving (5/8) × (4/9) = 20/72. Adding the two branches: 15/72 + 20/72 = 35/72. Trap 1/2 ignores the transfer and reads pouch Q as it stood at the start. Traps 5/9 and 4/9 each report a single branch as if the other could not happen.