This problem is courtesy of Mr. Lange’s friend Paul. The Queen of Hearts is a card game played at a local club. Participants purchase raffle tickets at $5 apiece, and one raffle ticket is chosen at random. The participant whose raffle ticket is selected will then select an envelope containing a card from a standard deck with two jokers. If the card selected is between a 2 and a 5 inclusive, the participant collects $20. If the card selected is between a 6 and a 9 inclusive, the participant collects $30. Any ace, 10, jack, or king pays $40, each joker pays $75, and the queens of diamonds, clubs, and spades each pay $100. The queen of hearts pays the jackpot, hence the game’s name. Suppose there are three cards remaining, the queen of hearts, the queen of diamonds, and the queen of clubs. Also suppose the jackpot pays $35000. Mr. Lange wishes to purchase 400 of the 1600 tickets sold in a given night. Answer the following questions: (a) What is the probability that Mr. Lange has the selected raffle ticket? Hint: Assume each ticket has an equally likely chance of being selected. (b) What is the probability that Mr. Lange selects the queen of hearts, given that he had the selected raffle ticket? Queen of diamonds? Queen of clubs? (c) Let X represent the amount Mr. Lange won. Construct the probability distribution for X. Hint: The amount won is equal to the amount paid to Mr. Lange less the cost of the raffle tickets. (d) Find E(X) and σX. Hint: First find E(X) and then use your answer to find Var(X).
This problem is courtesy of Mr. Lange’s friend Paul. The Queen of Hearts is a card game played at a local club. Participants purchase raffle tickets at $5 apiece, and one raffle ticket is chosen at random. The participant whose raffle ticket is selected will then select an envelope containing a card from a standard deck with two jokers. If the card selected is between a 2 and a 5 inclusive, the participant collects $20. If the card selected is between a 6 and a 9 inclusive, the participant collects $30. Any ace, 10, jack, or king pays $40, each joker pays $75, and the queens of diamonds, clubs, and spades each pay $100. The queen of hearts pays the jackpot, hence the game’s name. Suppose there are three cards remaining, the queen of hearts, the queen of diamonds, and the queen of clubs. Also suppose the jackpot pays $35000. Mr. Lange wishes to purchase 400 of the 1600 tickets sold in a given night. Answer the following questions: (a) What is the
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