You have two six-sided dice:

• One die is fair with the numbers 1 through to 6 occurring with equal probability
• The other die is loaded and always rolls an even number. It rolls a 2, 4 or 6 with equal probability.

You pull one die out of your pocket and roll it. You did not check which die it was before you rolled. (Assume you could have pulled either die out of your pocket with equal probability.)

a) The die shows a six. What is the probability that it is the loaded die? Calculate your answer using Bayes' rule. [1 mark]

b) You pull the other die out of your pocket and roll it. It shows a 5. What is the updated probability that the first die is the loaded die? Calculate your answer using Bayes' rule. [2 marks]