STA104/MTH135: Probability

Homework #3: Random Variables

Due: Friday, Sep 21

Text Problems are from Sheldon Ross, A First Course in Probability (6th/5th edn).

• Problems (pp. 173ff)
  

  7

  13

  17

  30

  32

  46

  49

  74

  (Be sure to explain your answer for 49 in detail)
• Exercises (pp. 184ff)

  3

  4

  20

• Another Problem
  One popular strategy for gambling on roulette, the martingale, works as follows, for a gambler who starts with a stake of $15. On US roulette wheels, Red and Black each appear with probability 18/38, while Green appears with probability 2/38.
  1. Bet $1 on Red. If Red appears (probability 18/38), quit with winnings $1. Otherwise,
  2. Bet $2 on Red. If Red appears (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 P[X>0].
  2. Are you convinced that the strategy is indeed a ``winning'' strategy? Explain your answer.
  3. Find E[X].

Good Luck!