Unexpectedly Intriguing!
April 21, 2005

Arnold Kling has been doing arithmetic in looking at the cost of risk versus it's benefit as it affects health care decisions, and when there's math to be done, we here at Political Calculations(TM) believe that there ought to be a tool to do it!

The tool below is based on the math described in Arnold's look at Costs, Benefits and Health Care, excerpted below:

Suppose that a medical test costs \$1000, and 98 percent of the time it fails to turn up anything that would affect treatment. The other 2 percent of the time, it results in a treatment choice that extends life by 5 years. How much does a year of life have to be worth in order for the test to have an expected value that exceeds its cost?

The answer is that the "expected number of life-years saved" is .02 times 5, or one-tenth of one year. If one year is worth more than \$10,000, then one-tenth of one year is worth more than \$1000, so that the test is worthwhile.

And now, here's the tool for performing the math, which I've tried to make more generally applicable for more decisions than just the health care problem above that provides its default values:

Cost and Benefit Data
Input Data Values
Cost of Taking Risk (\$USD)
Percentage Odds of Receiving Benefit (%)
Time that Benefit May Be Enjoyed (Years)
Value of Benefit per Year (\$USD)

Results and Decisions
Calculated Results Values
Value of the Benefit (\$USD)
Is the Risk Worthwhile?

Users of this calculator should pay close attention to the limitations inherent in the assumptions behind the math. For instance, can you really know how much a year of life is worth? Or have you fully accounted for the costs of the test - labor, materials, overhead, etc.? There's a lot of stuff that goes into the numbers you enter, and the usefulness of the tool will be proportionate to their quality....

### More Fun Uses

It does seem to me though that this tool is ideal for figuring out whether or not you should play the lottery. With that in mind, here is a table of odds for the Powerball lottery:

Powerball Lottery Odds
Matching Balls Prize Amount Odds
5 + Powerball Grand Prize 1 in 120,526,770.00
5 \$100,000 1 in 2,939,677.32
4 + Powerball \$5,000 1 in 502,194.88
4 \$100 1 in 12,248.66
3 + Powerball \$100 1 in 10,685.00
3 \$7 1 in 260.61
2 + Powerball \$7 1 in 696.85
1 + Powerball \$4 1 in 123.88
1 + Powerball \$3 1 in 70.39
The overall odds of winning a prize are 1 in 36.06.

For reference, here's an example of how to enter these odds in the tool above:

 "1 in 12,248.66" should be entered as: "100/12248.66"

This simple formula will convert the odds entered this way into the percentage format used by the calculator. For prizes other than the grand prize, enter "1" for the time that the benefit will be enjoyed, and the amount of the prize for the value of the benefit per year. For the grand prize, enter the number of years that the prize will be paid out for time, along with the amount that will be paid out each year to add up to the grand prize. Good luck!

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