Question 1: a) b) c) d) e) As the question stated that the first token selected is orange, the probability of selected the first token as orangeis 1.P (01) =1 The event B2/01 represents the probability of selecting the second token which is blue, given that the first token selected is orange. Since there are 30 tokens (21 oranges and 9 blue) After the first orange token is selected there are 29 tokens remaining (20 orange, 9 blue). Therefore: P (B2/01) = 9/29 The probability of selecting the orange token as the first and the blue token as the second is P (O1 and B2) - Orange token probability: P (0O1) =1 - Probability of blue token: P (B2/01) = 9/29 - Therefore: P(01and B2)=P (01) x P (B2/01) =1 x 9/29 = 9/29 The smallest number of blue tokens needed to achieve a probability of at least 0.11 for the event (O1 and B2) is 6. - Let the number of blue tokens be x - The total number of tokens is 21 + 27 + x =48 + x - The probability of (O1 and B2) is given by: - P (01 and B2) to be greater than or equal to 0.11, we need: - 1x(x/(48+x)=0.11 - Thereforex> 6 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