4. A player draws a card at random from a standard deck of 52, with each card equally likely to be chosen. After drawing

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4 A Player Draws A Card At Random From A Standard Deck Of 52 With Each Card Equally Likely To Be Chosen After Drawing 1 (98.54 KiB) Viewed 37 times

4. A player draws a card at random from a standard deck of 52, with each card equally likely to be chosen. After drawing a card and observing its value, it is returned to the deck and the deck is shuffled. The player wins $X where X is the number of cards he/she draws until the first Diamond is drawn. That is, if the first card is a Diamond, he/she wins $1. If the first card is not a Diamond, but the second is, he/she wins $2 etc. i) What is the distribution of X? ii) Write down E(X). The player is offered a financial option before the game. He/she can pay $Q in for the right (but not obligation) to swap his/her winnings for a fixed prize of $10. (The decision on whether or not to exercise this option is made at the end of the game.) iii) Let the player's eventual prize money be $M, assuming he/she exercises the option only if it increases his/her prize. Calculate the probability mass function of M. iv) Calculate the fair price of this option. That is, find Q such that E(X)=E(M)-Q.