STA104/MTH135: Probability

Homework #5: Discrete Random Variables

Text Problems are from Jim Pitman, Probability (1st edn).

• Exercises
  

  § 3.1:		6	13	16
  § 3.2:		1	2	6

• Another Problem (not optional)
  One popular strategy for gambling on roulette, the martingale, works as follows, for a gambler who starts with a stake of $15. In the US roulette wheels have 38 equally-likely outcomes, 18 each Red and Black and two Green (European wheels have 37 outcomes, with only one Green). The martingale strategy is:
  1. Bet $1 on Red. If Red appears (US probability 18/38), quit with winnings $1. Otherwise,
  2. Bet $2 on Red. If Red appears (US probability 18/38 again), quit with winnings -$1+$2=$1. Otherwise,
  3. Bet $4 on Red. If Red appears, quit with winnings -$1-$2+$4=$1. Otherwise,
  4. Bet $8 on Red. If Red appears, quit with winnings -$1-$2-$4+$8=$1. Otherwise,
  5. Quit anyway with total winnings -$1-$2-$4-$8=-$15. Hitchhike home.

  Let X denote the gambler's winnings (which could be negative!) when she quits.

  1. Find the probability distribution of X, i.e., find p(x)=P[X=x] for every real number x.
  2. Find P[X>0].
  3. Are you convinced that the strategy is indeed a ``winning'' strategy? Explain your answer.
  4. The queen of England is said to have a net worth of about 400 million dollars. If she commits to play the martingale strategy until she either wins $1, betting on Red every time and doubling her bet each time Red does not appear (as in steps 1-4 above, continuing indefinitely), or loses everything, what is the probability that she wins $1? What is the probability she loses $400,000,000? What is the expectation of her gain?

Good Luck!