Odds

• For an event E, let P(E) = n
• The Odds In Favor of E are n/(1-n) or n:1-n or n to 1-n
• The Odds against E are (1-n)/n or 1-n: n or 1-n to n
• Odds given in gambling are always odds against.
• Notice, Odds make it look more favorable then the probability.
• Odds against are also given as P(failure of E)/P(success of E)
• Probability from odds.
  • If the odds in favor of an event E are 3:2 then we know
  • P(E) = 3/n and P(not E) = 2/n
  • Further, we know that 3/n+2/n = 1
  • Thus 3 + 2 = n = 5
  • P(E)= 3/5, P(not E) = 2/5
• Do many problems from the book including 9, 13-16, 21, 37, 38, 39
• Roulette:
  • 38 numbered bins for a ball to bounce into
  • 1-36, 0 and 00
  • 18 red spaces, 18 black spaces, 2 green spaces
  • Three columns, 12 rows.
  • Bets on
    • Individual numbers
    • Colors
    • Even/Odd
    • Rows, or 4 rows
    • Columns
    • any 2 or 4 nearby numbers
    • Numbers less than, or greater than 18
  • The odds are given in terms of n to 1
  • The payout is
    • n-2 to 1 on the 37:1 odds
    • n-1 to 1 on any n:1 where n is an integer
    • round down(n) to 1 on any n:1 where n is not an integer

Expected Value

• What will happen over the long term?
• Let P_i be the probability of event i
• Let W_i be the profit/loss from event i
• E = P₁*W₁ + P₂*W₂ + ... + P_n*W_n
• The Fair price = expected value + cost to play
• Number 9 page 756
• 11, 13, 19, 21
• Careful on 25