Use the raffle information to answer the following questions.
|
Rule
of Thumb for Samples
Taken from a Large Population 1. Does the "Rule of Thumb for Samples Taken from a Large Population" apply to this raffle? |
There are 445 winners in this raffle. There are 40,000 tickets sold in this raffle. 445 / 40,000 = 0.011125. The sample is at most 5%. Independence can be assumed.
2. 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. Write
out all the decimal places.
| Event | x | Freq | P(x) | x * P(x) |
| $1,000,000 | $999,900 | 1 | 0.000025 | $24.9975 |
| $35,000 | $34,900 | 1 | 0.000025 | $0.8725 |
| $25,000 | $24,900 | 1 | 0.000025 | $0.6225 |
| $15,000 | $14,900 | 1 | 0.000025 | $0.3725 |
| $10,000 | $9,900 | 1 | 0.000025 | $0.2475 |
| $150 | $50 | 60 | 0.0015 | $0.075 |
| $130 | $30 | 60 | 0.0015 | $0.045 |
| $100 | $0 | 60 | 0.0015 | 0 |
| $90 | -$10 | 60 | 0.0015 | -$0.015 |
| $120 | $20 | 50 | 0.00125 | $0.025 |
| $110 | $10 | 50 | 0.00125 | $0.0125 |
| $90 | -$10 | 50 | 0.00125 | -$0.0125 |
| $80 | -$20 | 50 | 0.00125 | -$0.025 |
| 0 | -$100 | 39,555 | 0.988875 | -$98.8875 |
| Total | 40,000 | 1.000000 | -$71.67 |
3.
If all 40,000 tickets are sold, how much money will this charity make on their
raffle?
Revenue
-
Costs = Profit
Revenue = (40,000 tickets) *$100 / ticket = $4,000,000
Costs = Sum of all prizes
Costs = $1,000,000 + $35,000 + $25,000 + $15,000 +
$10,000 + 60 * $150 + 60 * $130 +
60 * $100 + 60 * $90 + 50 * $120 + 50 * $110 + 50 * $90 + 50 * $80 = $1,133,200
Profit = $4,000,000 - $1,133,200 = $2,866,800
4. Using the
prize money, not the random variable to define
an event, find:
A. P(Winning the $1,000,000 Dream
Home). 1
/ 40,000 = 0.000025 = 0.0025%
B. Odds against winning a Homedics Wireless Speakers and Ipod Deck. Losers : Winners 39,940 : 60
C. Odds in favor of winning a Samsung 4Gb Multimedia Player. Winners : Losers 50 : 39,950
D. P(Not Winning any prize).
There
are 445 prizes. There are 40,000 -
445 = 39,555 losers
39,555 / 40,000 =
98.8875%
E.
Odds in favor of winning any prize. There
are 445 prizes. There are 40,000 -
445 = 39,555 losers
Winners : Losers
445 : 39,555
F. P(Winning a prize worth at most $100).
There are 220 prizes worth $100 or less.
220 / 40,000 =
0.0055 = 0.55%
G. P(Not winning a prize worth more than
$100).
Hint:
Losing is included.
P(Not winning a prize worth more than
$100) = 1 -
P(Winning a prize worth more than
$100).
There are 225
prizes worth more than $100. 225 /
40,000 = 0.005625 = 0.5625%
1 - 0.005625
= 0.994375 =
99.4375%
H. Odds in favor of winning a prize
worth less than $120. There
are 270 prizes worth less than $120.
40,000
- 270
= 39,730 Winners
: Losers 270 : 39,730
I.
Odds against winning a prize at least $120. There
are 175 prizes worth $120 or more.
40,000 -
175 = 39,825 Losers : Winners
39,825 : 175
J. P(Losing money on the Raffle)
Hint: Losing or
winning a prize worth less than $100 is losing money.
There
are 39,715 people who either loss or won prizes worth less than $100.
39,715 / 40,000 =
0.992875 = 99.2875%.
K. Are "Winning a Homedics Wireless
Speakers and Ipod Deck" and "Winning a prize" independent events?
If
P(Winning a Homedics System) times P(Winning a prize) = P(Winning Homedics
System and Winning a prize)
Then the events are
independent.
P(Winning a Homedics
System) times
P(Winning a prize) = (60 / 40,000) (445 / 40,000) approximately 0.0000166875.
P(Winning a Homedics
System and
winning a prize) = P(Winning a Homedics System) = 60 / 40,000 approximately
0.0015.
The products are not
equal. The events are dependent.
L.
P(Winning a Homedics
Wireless Speakers and Ipod Deck.
| Winning a prize).
(60 / 40,000) / (445 / 40,000) = 60 / 445 approximately 0.134831461.
5. Find the
expected value of this raffle.
See Table. -$71.67
6. How is the
expected value related to the profit the charity makes on this raffle?
-$71.67
is the money loss, on an average, by each ticket buyer. This is the money gained
by the charity
for each ticket sale.
-$71.67
* 40,000
tickets = -$2,866,800
ticket
This is the loss by the ticket buyers and the profit earned by the charity.
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