Nov. 6, 1997
Perhaps this will be of interest to those seeking "realistic" problems
involving probability.
In the California State Lottery, there is a game called Decco. A brief
summary...
You pay $1 to play (to purchase a ticket).
You choose one card from each suit from a deck of 52 cards.
(For instance, Ace of Hearts, Ten of Clubs, Jack of Diamonds, Three of Spades.)
The State randomly chooses a card from each of the four suits.
If you match
° four of the cards, you win $5,000
° three of the cards, you win $50
° two of the cards, you win $5
° one of the cards, you get a free replay ticket.
My AP Statistics classes have analyzed the financial aspects of this game.
That is, they have calculated the amount of money the State "earns" for ever $1
spent to play this game. (It is advertised that 34% of the Lottery earnings
benefit California education). Without going into elaborate details, I'll just
summarize the findings.
° The number of possible 4-card combinations is 28,561.
° In 28,561 games, you would expect to
-match 4 cards 1 time.
-match 3 cards 48 times.
-match 2 cards 864 times.
-match 1 card 6,912 times.
-match 0 cards 20,736 times.
° In 28,561 games, the State would take in $28,561 and expect to have to pay
out winnings of $15,462. The expected State gain for 28,561 games is thus
$13,099, or about $0.4586 (approx. 46 cents) for each dollar spent.
What is interesting about Decco is the "Match 1 Card" event. When this event
occurs, you are given a free replay card. This is not equivalent to getting
your dollar back. Simply put, you don't get your dollar back and you
eventually end up in one of the other four categories (ie. Match 4,3,2, or 0).
Based on our calculations, the State takes in $6,912 for the 6,912 times this
is expected to happen in 28,561 games ... and the State expects to pay out
$3,742 during this 6,912 games. This represents a State gain of $3,170 for the
"Match 1 card" situation.
We (my class and I) wrote a little report on Decco, showing in some detail how
the figures above were obtained. (We believe our figures are correct, but
perhaps someone will find a flaw in our reasoning or calculations.) The report
also contains a picture of a Decco ticket, where the prizes and odds are
listed. It is not something that can be sent as an attachment. If anyone is
interested in seeing this 2-page report, you can send me a self-addressed
stamped envelope and I will be happy to send a copy to you. (Sanderson Smith,
Cate School, 1960 Cate Mesa Rd., Carpinteria, CA 93013).
Whether you approve of gambling or not, you can find some really interesting
probability problems associated with games of chance.