Odds

Discussion

• Odds are just another way of expressing probability.
• When we say odds, we generally mean odds against or the chances of the event not happening.
• The odds against are computed as follows:

•                         P(Event fails to occur)
  odds against an event = -----------------------
                              P(event occurs)
      

• Odds are usually expressed as n:m or n to m
• Example 1

  An 8 sided dice is tossed, what are the odds against
      Rolling a 2
  
          P(2) = 1/8
          P(not 2) = 7/8
                              7/8
          Odds against a 2 = ----- = 7/8 × 8/1 = 7/1
  	                    1/8
  
          or 7:1 or even 7 to 1
  
       Rolling a number less than 5
           P(n<5) = 4/8 = 1/2
  	 P(n>=5) = 4/8
  	                 1/2
  	 Odds against = ----- = 1 to 1 or even odds.
  	                 1/2
      

• The odds in favor of an event are computed much the same way

•                      P(E)
  odds in favor of E=-------- 
                      1-P(E)
      

• Example 2:

  Two six sided dice are rolled.  
  Find the probability for any given number to be the sum
  
  There are 36 ways the dice can be rolled
  
       1: can't happen                 P(1) = 0
       2: Both must be 1 1-1           P(2) = 1/36
       3: 1-2, 2-1                     P(3) = 2/36
       4: 1-3, 2-2, 3-1                P(4) = 3/36
       5: 1-4, 2-3, 3-2, 4-1           P(5) = 4/36
       6: 1-5, 2-4, 3-3, 4-2, 5-1      P(6) = 5/36
       7: 1-6, 6-1, 2-5, 5-2, 3-4, 4-3 P(7) = 6/36
       8: 2-6, 3-5, 4-4, 5-3, 6,2      P(8) = 5/36
       9: 3-6, 4-5, 5-4, 6-3           P(9) = 4/36
       10: 4-6, 5-5, 6-4               P(10) = 3/36
       11: 5-6, 6-5                    P(11) = 2/36
       12: 6-6                         P(12) = 1/36
  
      What is the chance of rolling a 7  or 11?
  
      P(7 or 11) = 6/36+2/36 = 8/36  (or 22%)
  
      What are the chances of rolling something other than a 7 or 11
  
      1-P(7 or 11) = 36/36 - 8/36 = 28/36 (or 78%)
  
      What are the odds against rolling a 7 or 11
  
            28/36 ÷  8/36 = 28/8 or 28 to 8
  
      What are the odds in favor of rolling a 7 or 11
  
            8/36 ÷ 28/36 = 8/28 or 8 to 28
      

• Example 3, problem 36, page 676

  The odds in favor of an event are 2:7.  
  Find the probability of the event occurring.
  
  P(E)/P(!E) = 2/7
  P(E) + P(!E) = 2/x + 7/x= 1
               = 9/x = 1
  	     x = 9
  
  P(E) =  2/9 or .22, 22%
      

Additional Problems

• 16
• 20
• 26
• 30
• 38

Homework