Proj: Raffle Stat. Answers

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

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

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.

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.