Math CentralQuandaries & Queries


Question from jp, a student:

you pay $.50 and pick a four-digit number. The state chooses a four-digit number at random and pays you $2500.00 if your number i chosen. What are the expected winnings from a $.50 Pick 4 wager?

Hi JP. The quick answer is that you can't "expect" any winnings at all!

I understand your question though, to mean how much money gets paid back to the gamblers under this arrangement.

If all four-digit numbers are equally likely, then the chance a ticket's number matches the winner is 1 in 10 000. Each digit is independent of the others, and presuming that leading zeroes are permissible (for example, I can pick the number 0007 if I want), then each digit has 10 possibilities. So 10 choices for each digit means there are 10 x 10 x 10 x 10 = 10 000 possible four-digit numbers to choose.

If each ticket is $0.50, then typically you would require $5000 to cover the whole range. If you did that, you would be guaranteed to receive $2500. So you'd be losing half your money.

This is how lotteries operate. They provide a chance at a large win and profit by having many losers of small sums. This kind of ticketing arrangement you have described would likely yield 100% profit (less once printing and administration costs are included) since for every $5000 in tickets, the lottery only pays out $2500.

Of course, if there are very few gamblers involved and consequently few tickets, things become less predictable. If all the gamblers choose the same number (say 1234) and that wasn't the random winning number, the lottery gets to keep everything and pay out no winnings at all. But if the winning number was 1234, the lottery would go broke. That's why gambling organizations including lotteries are extremely dependent on having a large number of independent low-stakes gamblers rather than just a few high-stakes gamblers. It makes for a predictable revenue return.

Stephen La Rocque.



Probabilists calculate the expected value by looking at each possible outcome, multiplying by the probability of that outcome, and adding all these products. For your experiment there are two possible outcomes, you win $2500.00 or you win $0.00. Multiply each outcome by the probability of that outcome and then add the two terms. Your result should be exactly what Stephen obtained. You expect to loose half your money.


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