1. Suppose that a friend gives us the chance to participate in a game. The game is as follows. There is a bucket containing 30 blue tokens, 21 orange tokens and 27 red tokens. We are to blindly reach into the bucket and randomly choose one token. Keeping this token (i.e. the chosen token is not returned to the bucket) we are to once again reach in blindly and randomly retrieve another token. If the first token we have chosen is orange and the second is blue then we win the game. Otherwise we lose. Define the two events below as follows: e (); = first token selected is orange. e B, = second token selected is blue. (a) What is P(0O4)? (4 marks) (b) In words, explain what the event B3|O; is. (4 marks) () What is P(B3|041)? (4 marks) (d) What is P(O; and Bs)? Show your workings. (4 marks) (e) Suppose that you would like to increase the probability that you win the game where you propose the same number of orange tokens (21 orange tokens) and red tokens (27 red tokens) to be used but with a new total number of blue tokens. You would like your probability of winning the game to be at least 0.11 (i.e. P(O1 and B2) > 0.11). Use Excel to calculate the smallest number of blue tokens needed to achieve this? What is the smallest number of blue tokens needed to achieve this? Explain very clearly how you obtained your answer. (4 marks) NOTE: You must be very clear how you obtained your answer. That is, whoever reads your answer must understand fully how you obtained it and be convinced that you are confident that your answer is correct