3. There is an old saying, "Try, try, try again" it alludes to the sentiment that success may not happen on the first iteration of an attempt. Suppose a teenager is trying to re-create a magic trick he witnessed, however, since he doesn't know how the trick was performed he will rely on randomness and probability to get the job done. Let's suppose he has an abbreviated deck of cards containing all regular face cards and aces but that only has the even numbered cards, a total of 36 cards. His "trick" revolves around a story to which the punch line suggests that he is either: a King, will marry a Queen, is an Ace, or the Jack-of-all-trades. Basically, he shuffles the deck of 36 cards then select one, hoping that it is one of the sixteen favourable cards (any: King, Queen, Ace or Jack). Understanding the odds of selecting one of the desired cards to be low, he has used his knowledge of probability to come up with a really good way of having this "trick" workout. He has prepared three hilarious excuses, to entertain the crowd if he gets a card he doesn't intend on seeing. Then he re-assures the crowd he can be trusted by returning the unsuccessful card to the deck then re-shuffling and tries again. If his hilarious excuses run out he does the best he can to make up something on the spot and continues his process until he gets his success. Let X be the number of tries he takes to get this "trick" right. Meaning X~geom (p = =). For all parts of this question if rounding use at least 4 digits after the decimal. a) What is the probability that he will complete the trick successfully using at most 1 hilarious excuse? [1] Answer: b) While practicing this trick, he convinces a friend to give him $500 dollars if he can get the trick to work without using any excuses, $25 dollars for one excuse used, $1.25 dollars for two excuses used etc. The money he would get from this bet is a function of X given by 500[0.05*-1]. What amount of money does he expect to win? [1]