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Early on the morning of October 15, 1997, the robot spaceship Cassini was launched successfully from Cape Canaveral. The ship sped away toward Venus, its first destination in a looping seven-year voyage designed to bring it to Saturn on July 1, 2004. Its departure ended an acrimonious debate over whether it should be launched, carrying as it did some 72 pounds of highly toxic fuel containing radioactive plutonium 238.
This article takes no stand in the controversy; it's concerned with the mathematical quality of the debate. It's difficult to compute the probabilities of very unlikely events, and after the calculations are done it's difficult to compare and evaluate the results. This leads to poor communication, and after a while everyone has retreated to extreme positions. Just before Cassini launched, the National Space Society (in favor of the launching) was quoting odds of 1.3 million to 1 there would be no release of plutonium, while the opponents were quoting odds of 138 to 1: almost 10,000 times greater. The truth, we assume, was somewhere in the vast gap between these two figures.
One of the important things mathematicians do is build tools for people to use in thinking about numbers. Risk comparison is an area where tools are needed, because of the trouble everyone has dealing with numbers much smaller than our normal experience. Is there a real difference between "one in a million" and "one in a billion"? Indeed there is! One number is a thousand times the other, yet both are much too small for us to deal with effectively.
Tools exist to think about and compare such tiny quantities. The best of them is the logarithm, presented to us by John Napier, a Scottish nobleman, in 1614. Members of the general public probably don't know or don't remember what a logarithm is, but everyone has encountered logarithmic scales of measurement. Four common examples are:
All of these scales convert very small numbers (the tiny amount of light from a star, the imperceptible wiggle under a seismograph, the minute pressure of a sound wave, the concentration of hydrogen atoms in moles per liter) into familiar numbers we deal with all the time.
We can do the same for risk. Let's define the risk magnitude of a possible event to be
where p is the probability of the event. Very often the probability is stated as "1 in N" where N is some large number. In that case the risk magnitude of a "1 in N' event is
An event certain to happen has probability 1 and , since log 1 = 0, risk magnitude 10. An event that occurs 1 in a million times has p = 10-6 risk magnitude 10 - 6 = 4. An event of risk magnitude 0 has probability 10-10 (1 chance in 10 billion). An event even less likely than that has negative risk magnitude.
Just as an earthquake of magnitude 7.5 is 10 times as powerful as one of magnitude 6.5, increasing risk magnitude by 1.0 makes an event 10 times as likely to happen. An event with risk magnitude 6.5 is 10·10 = 100 times as likely as an event of risk magnitude 4.5.
Using the risk magnitude scale, we can quickly get a handle on how large a risk is. For example, you may remember the excitement over the flawed Pentium chips back in 1994. Everyone with a flawed chip felt they had a new one. However, tests showed that the risk magnitude of getting an erroneous calculation out of the chip was only 0.1.
In the case of Cassini, of course, the purported risk magnitudes were all over the scale. The opponents were claiming a risk magnitude of 10 - log 138 = 7.9, while the proponents were claiming 10 - log (1.3 million) = 3.9. When this kind of disparity arises, it's clear the two sides are not talking to each other and all the numbers should be viewed with some suspicion.
It might be fun for a class to monitor the newspapers for claims and statements of risk and keep a chart of the associated risk magnitudes. (The RM values can be computed using any scientific calculator. It's not even necessary for students to understand what a logarithm is, as long as they understand that a difference of 1 in RM means a difference of 10 times in the risk.) Here's a chart with some fairly well-established risk magnitudes:
Event
Risk Magnitude
The Sun will rise tomorrow
10.0
The next child born in your family will be a girl (or a boy, if you prefer)
9.7
A major hurricane strikes the North Carolina coast in a given year
9.1
An asteroid at least 100 meters in diameter will collide with the Earth next year
8.0
A person is involved in a car accident while making a 10-mile trip
5.9
A comet will collide with the Earth next year
3.5
You buy one ticket and it wins the Lotto in Virginia
3.2
A person dies in a car crash while making a 1000-mile trip
2.3
A person dies in a plane crash while making a 1000-mile flight
0.9
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Posted November 1, 1997. Features remain online as long as they remain current; they may be updated if new information becomes available.
Copyright © 1997, Center for Mathematics and Science Education. Teachers have permission to duplicate this page for use in teaching their own classes. All other rights reserved. You are welcome to link to this page, but do not copy its contents.
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