Question 1 [3 marks] 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 a 5 or a 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? [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 rst die is the loaded die? [2 marks] Question 2 [4 marks] You know the following statistics about re: - The probability of your house catching re on any particular day is 1 in 10,000 - Your re alarm correctly detects a house re 95% of the time - The probability that your re alarm sounds on a day when there is no re [a false alarm) is 1 in 100. a) Your alarm goes off. What is the probability that your house is on re? [1 mark] b) Many people given this problem estlmate the probability of the house being on re as close to 95%. Provide one possible explanation for this error. [1 mark] c) Express and solve this problem using natural frequencies. [2 marks]