Now, let’s consider martingale and Markov processes in a standard Blackjack game. Cards are dealt one at a time from a standard 52-card randomly shuffled deck, and points are awarded to the lone recipient based on the number on the card (2 to 10) or 11 if the dealt card is a face card or Ace. Let St represent the number of points to be held by the recipient after t cards have been dealt by the dealer. For parts a through c, suppose that the cards have been dealt without replacement. For parts d, e, and f, assume that the cards have been dealt with replacement and that 1 point is awarded if the number on the card is a 2 through a 6, 0 points are awarded if the number on the card is 7, 8, or 9, and -1 point is awarded if the card is a 10, a face card, or an ace. We note that this is the most common point system used by card counters playing blackjack in casinos.

a. Is this process stochastic?

b. Is this process Markov? This part of the problem is a little more difficult.

c. Is this process a submartingale?

d. Is this process (with replacement) Markov?

e. Is this process (with replacement) a submartingale?

f. Is this process (with replacement) a martingale?