Hey i need help with this assessment please. I must achieve near full marks if possible!

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. 5 = second token selected is blue. (a) What is P(0,)? (b) In words, explain what the event B|(); is. () What is P(B2]01)? (d) What is P(( and B2)? Show your workings. (4 marks) (4 marks) (4 marks) (4 marks) (e] Suppose that you would like to increase the probability that yon 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(O and B3} = (.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 yvou obtained yvour answer. (4 marks) NOTE: You must be very clear how you obtained vour answer. That is, whoever reads your answer must understand fully how vou obtained it and be convinced that you are confident that vour answer is correct. 2. Consider a standard 52 card deck of playing cards. In total there are four cards that are Aces, four cards that are Kings, four cards that are Queens and four cards that are Jacks. The remaining 36 cards are four each of the numbers 2, 3, ..., 10. That is there are four cards that are twos, four cards that are threes ete. For this question, suppose that we reduce the mumber of cards in the deck by o removing two of the Kings o removing three other cards that are not Kings The cards that are removed are discarded and are not used for the remainder of this question. As such we now have a deck that consists of just 47 cards. Suppose that a card is randomly drawn from this reduced sized deck. Let K, denote the event that this card is a King. This card that was drawn from the deck of cards is now discarded and we continue with a deck of just 46 cards. Suppose that a second card is now randomly drawn from this 46-card deck and let K3 denote the event that this card is a King. This card that was drawn from the deck of cards is now discarded and we continue with a deck of just 45 cards. Suppose that a third card is now randomly drawn from this 45-card deck and let Ky denote the event that this card is a King. Answer the following questions. (a) What is P{K;)? (4 marks) (b) Given that the first card drawn was a King, what is the probability that the second card drawn is a King? That is, using our notation, what is P(K3|Ky)? (4 marks) (c) What is the probability that the first card drawn is a King and the second card drawn is a King? That is, what is P(K; and K)? Show your workings. (4 marks) (d) Given that the first card drawn is a King and the second card drawn is a King, what is the probability that the third card drawn is not a King? That is, what is P {Kff [{K1 and K3})? (4 marks) {e) What is the probability that the first two cards drawn are Kings and the third card drawn is not a King? That is, what is P(K; and K5 and K)? Show your workings. (4 marks)