A fair die is rolled. If the score is 1, 2, 3 or 4, bag A A\,A is selected; otherwise bag B B\,B is selected. Bag A A\,A contains 2 red and 3 blue counters. Bag B B\,B contains 4 red and 1 blue counter. A counter is drawn, replaced, and then a second counter is drawn from the selected bag.
Find the probability that both counters are red.
Given that both counters are red, find the probability that bag B B\,B was selected.
Determine whether the colours of the two counters are independent before the bag is known.
200 exam-style questions on OCR (MEI) A Level Maths 2.3 Probability, covering 2.3.1 Calculate the probability of an event, 2.3.2 Complementary events, 2.3.3 Expected frequency of an event, 2.3.4 Diagrams to calculate probabilities, 2.3.5 Mutually exclusive and independent events, 2.3.6 Add probabilities for mutually exclusive events, 2.3.7 Multiply probabilities for independent events, 2.3.8 Mutually exclusive and independent events (notation) (A-level only), 2.3.9 Venn diagrams for probabilities (A-level only), 2.3.10 Conditional probabilities (A-level only), and 2.3.11 Independence via conditional probability (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.