Expected Value

• Expected Value tells a way to find what will happen in the long run in a situation involving profit or loss.
• Expected value is calculated by
  1. Multiplying the probability of each situation by its profit or loss.
  2. Example: You pay $2 to play a dice game. If you roll a 6 you win $12, otherwise you lose. Find the expected value.

     The probability of a win is 1/6
     The probability of a loss is 5/6
     
     You win $10, otherwise you lose $2
     
     E = 10*1/6 - 2 * 5/6    = 10/6 - 10/6 = 0
     This is a fair game.
     	

  3. In this example, you can expect to stay at even money if you play the game a long time.
• How would the expected value change if the payoff was $11, $13?

  E11 = 9/6-10/6 = -1/6 = $ -0.17  
  E13 = 11/6-10/6 = 1/6 = $ 0.17
      

• What is the expected value when tossing a 4 sided die?

      P(1) = 1/4
      P(2) = 1/4
      P(3) = 1/4
      P(4) = 1/4
  
      E = 1 * 1/4 + 2 * 1/4 + 3 * 1/4 + 4 * 1/4
        = (1+2+3+4) / 4 = 10/4 = 2.5
      

• What is the expected value of the sum of two dice?

      
  
      
  Value
  Probability
      
   2 
   1/36
      
   3 
   2/36
      
   4 
   3/36
      
   5 
   4/36
      
   6 
   5/36
      
   7 
   6/36
      
   8 
   5/36
      
   9 
   4/36
      
  10 
   3/36
      
  11 
   2/36
      
  12 
   1/36
      
  
      So the expected value is 
          2*1/36 + 3*2/36 + 4*3/36 + 5*4/36 + 6*5/36 + 7*6/36
        + 8*5/36 + 9*4/36 + 10*3/36 + 11*2/36 + 12*1/36
  
        = (2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12)/ 36
        = 252/36 = 7
      

• Problem 2, page 620
• Problem 4, page 260
• Problem 16, page 621

Homework