1. A hand of 5 cards contains 2 red cards and 3 black cards. Trish plays the following game: A card is drawn from the hand. If the card is red, the game stops immediately. If the card is black, this black card is set aside and a red card is put into the hand in its place. Then another card is drawn from the hand and the same process is repeated. The game continues until Trish draws a red card ( Note that there are always 5 cards in the hand before Trish draws a card.)
Let X be the number of black cards Trish drew during the game. (That is, the value of X is determined after the game has finished.)
(a) Make a table showing the probability distribution function(pdf) and cumulative distribution function (cdf) for X. Hint: draw a probability tree modeling the game.
(b) If it is known that Trish drew at least 1 black card, what is the probability that she drew no more that 2 black cards? Hint: Use your table from part (a)
(c) If Trish pays $2 to play this game, and receives $2 for each black card she draws, how much should she expect to win or lose, on average, when she plays this game?
(d) Find sigma (X).