Odds

• Odds are another way to measure probability.
• Odds are either "in favor of an event" or "against an event"
• The Odds in favor of an even E
  • If P(E) = p
  • P(not E) = 1-p
  • The odds in favor of E are P(E)/P(note E) (or p/(1-p))
• What are the odds in favor of rolling a 5 on a 6 sided die
  • P(E) = 1/6, P(Not E) = 5/6
  • Odds in favor are 1/5
• We normally don't express odds as a fraction
  • 1 to 5, 1:5, 1-5, or 1 in 5 are much more common
  • But we do reduce the fraction.
• Odds against E
  • This is what is used in games of chance.
  • Odds against E are P(not E)/ P(E)
• Given the odds, can you find the probability of E?
  • The odds against E are 3:5.
  • The denominator for the probabilities are 3+5
  • P(not E) = 3/8, P(E) = 5/8
• P 746 do some problems.

Expected value

• Expected value is used to determine the success (profit or loss) of a venture of time.
• This involves an event or set of events that cover the entire space
• i.e. Σ P(E_i) = 1
• It also includes a profit or loss for each event A_i
• For each event,
  • Find P_i
  • Find A_i
  • Compute P_i × A_i
• Add up all of the computed values.
• The fair price is the result to be paid for an expected value of 0
• fair price = expected value + cost to play.