A bag contains 5 red beads and 7 blue beads. Two beads are removed at random without replacement. Workout the probability that the two beads are the same colour.

This question is most simply solved with a probability tree diagram, where you just follow the paths of picking two same coloured beads. The first branch would be picking two red beads. P(R1) = 5 / 12, and P(R2) = 4 / 11, as the red bead would have been removed. The second branch would be picking two blue beads, where P(B1) = 7 / 12 and P(B2) = 6 / 11 for the same reason.
Our total probability is then (R1 x R2) + (B1 x B2) = (5 / 12 x 4 / 11) + (7 / 12 x 6 / 11) = 31 / 66