Project: Raffle

Use the raffle information to answer the following questions.  
1.  Make a probability distribution table for this raffle.  Use the prize money as the events.  
     Remember to subtract the price of the ticket from the events when determining the 
     random variables.

Event x Freq P(x) x  *  P(x)
$1,000,000 $999,900 1 0.00002 19.998
$50,000 $49,900 1 0.00002 0.998
$25,000 $24,900 1 0.00002 0.498
$15,000 $14,900 1 0.00002 0.298
$10,000 $9,900 1 0.00002 0.198
$500 $400 140 0.00280 1.120
$200 $100 635 0.0127 1.270
$100 0 635 0.0127 0
0 -$100 48,585 0.9717 -$97.170
Total   50,000   -$72.79

2.  If all 50,000 tickets are sold, how much money will this charity make on their raffle?
     Revenue - Costs = Profit
     Revenue = (50,000 tickets) *$100 / ticket = $5,000,000

     Costs = Sum of all prizes
     Costs = $1,000,000 + $50,000 + $25,000 + $15,000 + $10,000 + 140 * $500 + 635 * $200 + 
                   635 * $100 = $1,360,500

      Profit = $5,000,000 - $1,360,500 = $3,639,500   

3.  Using the prize money, not the random variable to define
     an event, find:
     A. P(Winning the $1,000,000 Dream Home) 
1 / 50,000 

     B. Odds against winning the $1,000,000 Dream HomeLosers : Winners     49,999 : 1

     C. Odds in favor of winning $50,000 or a Mercedes-Benz or a Cadillac.  Winners : Losers     1 : 49,999

     D. P(Not Winning something) There are 1,415 prizes.  There are 50,000 - 1,415 = 48,485 losers
         48,485 / 50,000

     E. Odds in favor of winning any prize.   There are 1,415 prizes.  There are 50,000 - 1,415 = 48,485 losers
         Winners : Losers     1,415 : 48,485 

     F. P(Winning a prize worth at most $500).
         P($100) + P(200) + P($500) = (635 + 635 + 140) / 50,000 = 1,410 / 50,000

     G. P(Winning a prize worth more than $500).
         Number of prizes worth more than $100 is 5.     5 / 50,000   

     H.  P(Not winning a prize worth less than $200). 
          Number of prizes not worth less than $200 are
635 + 140 + 5 = 780.      780 / 50,000

     I.  Odds against winning at a prize worth at least $200.
         Number of prizes worth at least $200 are 635 + 140 + 5 = 780.
         Number of prizes not worth at least $200 are 50,000
- 780 = 49,220
 
        Not Worth $200 : At Least Worth $200     49,220 : 780

     J.  Are "winning $500" and "winning a prize" independent events?
          If P(Winning $500) times P(Winning a prize) = P(Winning $500 and winning a prize)
          Then the events are independent.
          P(Winning $500) times P(Winning a prize) = (140 / 50,000) (1,415 / 50,000) approximately 0.00007924.
          P(Winning $500 and winning a prize) = P(Winning $500) = 140 / 50,000 approximately 0.0028.
          The products are not equal.  The events are dependent. 

     K.  P(Winning $500 | Won a prize)
          P(Winning $500 and winning a prize) / P(Winning a prize)
          (140 / 50,000) / (1,415 / 50,000) = 140 / 1,415 approximately 0.0989

     L.  Are "winning the $1,000,000 dream home" and "winning a prize" independent events?
         
If P(Winning $1,000,000) times P(Winning a prize) = P(Winning $1,000,000 and winning a prize)
          Then the events are independent.
          P(Winning $1,000,000) times P(Winning a prize) = (1 / 50,000) (1,415 / 50,000) approximately 0.000000566.
          P(Winning $1,000,000 and winning a prize) = P(Winning $1,000,000) = 1 / 50,000 approximately 0.00002.
          The products are not equal.  The events are dependent

     M.  P(Winning the $1,000,000 Dream Home | Won a prize
         
 P(Winning $1,000,000 and winning a prize) / P(Winning a prize)
          (1 / 50,000) / (1,415 / 50,000) = 1 / 1,415 approximately 0.000707.
   

4.  Find the expected value of this raffle.
     See Table.  -$72.79

5.  How is the expected value related to the profit the charity makes on this raffle?
    
-$72.79 is the money loss, on an average, by each ticket buyer.  This is the money gained by the charity
    for each ticket sale.

     -$72.79 * 50,000 tickets = -$3,639,500
      ticket 

      This is the loss by the ticket buyers and the profit earned by the charity. 

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