Question

2. Counting cards in blackjack (as seen in movies such as Rainman, The Hangover and 21) consists in keeping track of a count: the higher the count, the more likely it is for the next card drawn to be a high card. One version of this uses the following setup: . the cards 2, 3, 4, 5, 6 are given the value +1, e the cards 7, 8, 9 are given the value 0, e the cards 10, J, Q, K, A are given the value -1. Therefore, if many low cards are drawn, then the count will increase, indicating that the concentration of high cards left in the deck is getting higher. Consider playing blackjack with one standard deck of cards (52 cards). Define X to be the count after drawing out two cards randomly (a) What are the possible values of the random variable X? (b) Find the probability distribution of X (c) Plot the probability distribution as a line graph and plot the cumulative probability distritution as a step function. (d) Calculate the expected value E(X) and the variance V(X) (e) In casinos, blackjack is often played with 6 or 8 decks shuffled together (or even with continuous shuffling ma- chines). Suppose now that you play with a large (infinite number of decks all shuffled together. What is the prob- ability distribution of X in that case? (f) If we were to calculate the expected value and variance for the distribution found in part (e), would we get the same answers as in (d)? If not, would we get a bigger/smaller answer? Justify without calculating E(X) and V(X) ex- plicitly

Answer 1

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