Suppose that a friend gives us the chance to participate in a game. The game is as follows. There is a bucket containing 24 green tokens and 16 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 green and the second is red then we win the game. Otherwise we lose. Define the two events below as follows:

G1 = first token selected is green.

R2 = second token selected is red.

(a) What is P(G1)?

(b) In words, explain what the event R2|G1 is.

(c) What is P(R2|G1)?

(d) What is P(G1 and R2)? Show your workings.

(e) Suppose that you would like to increase the probability that you win the game where you propose the same number of green tokens to be used (24 green tokens) but with a new total number of red tokens to be used. You would like your probability of winning the game to be at least 0.25 (i.e. P(G1 and R2) 0.25). What is the smallest total number of tokens needed (number of green tokens plus the number of red tokens) to achieve this? For this total number of tokens, how many are red? Explain very clearly how you obtained your answer.

NOTE: there are easy ways and hard ways to tackle this question. If you wish you may use Excel to help you. We do not mind how you find your answer (provided you arrive at the answer yourself). However, you must be very clear how you obtained your answer. That is, whoever reads your answer must understand fully how you obtained your answer and be convinced that you are confident that your answer is correct.