STA230/MTH230: Probability

Homework #5: Discrete Random Variables

Text Problems are from Jim Pitman, Probability.

Note:This short homework set is due Monday, so it won't interfere as much with the midterm on Wednesday.

• Exercises
  

  § 3.1:		6	13#	16
  § 3.2:		1	2*	6

  # Balls are removed, examined, and replaced one at a time
  * The first and second lists are both shuffled in random order
• Another Problem (not optional)
  One popular strategy for gambling on roulette, the "martingale," gives a very high probability of a positive outcome. Here's how it works, 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 to double the bet after each loss:
  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. You're broke, so you have to 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 (in 2018) of about 500 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 all $500,000,000? What is the expectation of her net gain?

Good Luck on the Midterm! Enjoy Fall Break!