Combined Conditional Probabilities (Cambridge (CIE) IGCSE International Maths)

If a question says "two cards are drawn" then you may assume that they draw 1 card followed by another card without replacement (the maths is the same).

Try not to simplify your probabilities too early as it is easier to add probabilities together when they all have the same denominator!

A bag contains 10 yellow beads, 6 blue beads and 4 green beads.

A bead is taken at random from the bag and not replaced.

A second bead is then taken at random from the bag.

(a) Find the probability that both beads are different colours.

Let Y, B and G represent choosing a yellow, blue and green bead

List all the possibilities of different colours

Remember that YB (yellow first, then blue) is different to BY (blue first, then yellow)

YB, BY, YG, GY, BG, GB

Use the "or" rule to add the cases together

P(different colours) = P(YB) + P(BY) + P(YG) + P(GY) + P(BG) + P(GB)

Calculate each conditional probability separately, remembering the number of beads changes after one is drawn and not replaced

For example, P(YB) =

Multiply the pairs of fractions together and add their results

Simplify the answer

The second bead is not replaced and a third bead is taken at random from the bag.

(b) Find the probability that all three beads are the same colour.

List the possibilities

YYY, BBB, GGG

Use the "or" rule to add between cases

P(all the same colour) = P(YYY) + P(BBB) + P(GGG)

Use conditional probabilities in each separate case, remembering the number of beads changes after each one is drawn and not replaced

Multiply the triplets of fractions together then add their results

Simplify the answer