4. 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).
This shows how to make a table with probability distribution function and cumulative distribution function by drawing a probability tree, calculate the probability for a given situation, and calculate how much is expected to be won or lost.