In an Oregon Lottery game called Cash Crossword, a player purchases a card for t

Question

In an Oregon Lottery game called Cash Crossword, a player purchases a card for two dollars. Scratching the card reveals a crossword puzzle with a certain number of words that have been circled. Or, no words have been circled. The player wins, if there are one or more words that have been circled. The more words that have been circled, the more the player will win.

Assume that a total of 2,597,163 scratch-it cards are printed. Among these, the table below shows the number of winning cards printed for each possible number of circled words and the amount that the player can win for each of those winning cards. (All cards are distributed to outlets. No cards are printed with only one or two words circled.)

Each card can only win in a single category. That is, if a card contains ten circled words, the player can’t claim a win for 10 words as well as any smaller number of words.

Circled Words

10

9

8

7

6

5

4

3

Number of Cards

5

12

445

5200

39000

39000

273000

377000

Winnings

$15000

$1000

$100

$50

$20

$10

$4

$2

a. Build a probability distribution table that gives the probability of winning for each different amount that can be won. (Be sure and include the probability that a player may win nothing.)

b. On average, what is the net amount that the player can expect to win or lose after purchasing a single card?

c. If on average, 37% of the scratch-it cards distributed are purchased, how much can the Oregon Lottery Commission expect to gain or lose from those sales? Do not consider administrative costs, like overhead, printing costs, distribution costs, etc.

d.How many non-winning scratch-it cards have been printed and distributed?

e. Assuming the same number of winning cards are printed, how many non-winning cards would need to be printed and distributed to make this game break-even for the lottery commission. (In this case, assume that all cards printed are sold. Do not consider administrative costs, like overhead, printing costs, distribution costs, etc.)

Circled Words

10

9

8

7

6

5

4

3

Number of Cards

5

12

445

5200

39000

39000

273000

377000

Winnings

$15000

$1000

$100

$50

$20

$10

$4

$2

Explanation / Answer

Thus, average loss = 2 - 1.2319

= 0.7680  dollars is the average loss amount

Thus,

Average gain for oregon state lottery = 0.7680 * 2,597,163 * 37%

= 738,085 dollars

Total of 1967501 non-winning cards have been printed. Assuming, 37% of them got sold as well,

then the non-winning cards sold = 1,967,501 * 0.37

= 727975 cards

To break even, the number of non-winning lottery cards printed must be equal to 970,088. Thus,

total number of cards to be printed = 177,750

The calculations are as follows:

Hope this helps.

Words Cards probability Amount Expected value 10 5 1.92518E-06 15000 0.028877664 9 12 4.62043E-06 1000 0.004620426 8 445 0.000171341 100 0.017134081 7 5200 0.002002185 50 0.100109235 6 39000 0.015016385 20 0.300327704 5 39000 0.015016385 10 0.150163852 4 273000 0.105114696 4 0.420458785 3 273000 0.105114696 2 0.210229393 Rest 1967501 0.757557766 0 0 Sum 1.231921139

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