## What really has a 1 in a million chance?

This is part of a topic that's fun to do in class. First I ask students

If you overheard the phrase "1 in a million chance" in someone else's casual conversation, what might they be talking about?

and students typically offer both iconic examples (xxx cross-ref: winning the lottery, struck by lightning) and more imaginative suggestions. Then I ask

How could we get data on actual casual usage of the phrase "1 in a million chance"?

and neither the students nor I can think of anything much more practical than searching in blogs, so I show those results. Finally I ask for suggestions for events that we can convince ourselves really do have a 1 in a million chance (up to a factor of 2, let's say). Then I go through the students' suggestions; can we quantify the chances, and (if so) are they around 1 in a million?

## Example and nonexamples

The "bullets" are the examples, with YES or NO indicating whether the "1 in a million" chance estimate is reasonable.

Let's first dispose of trite examples from games of chance or sampling

• 20 coin tosses (by me) all coming up Tails. YES
• Winning the current California State lotto game if you buy 40 tickets. YES
• specify three famous people; getting one of these people on the Wikipedia "random article" link. YES
If you tossed the coins then the answer would be NO, unless I'm very confident you lack the ability to fool me ....

One interesting example is

• A major earthquake on the Hayward fault in the next 50 minutes. YES
Comment: A 2007 estimate puts the chance of a major (> 6.7 magnitude) earthquake on the Hayward fault at about 1% per year. Since my classroom is a few hundred yards from the faultline and classes are often 50 minutes, the numbers work out nicely!
Lest others become complacent, one can add e.g.

Another example:

• One of the next 25 babies born in the U.S. will become President. YES
Comment: The U.S. birth rate is currently about 4.3 million per year. If we guess a President will serve on average about 6 years, then 1 in 6 times 4.3 million = 25 million babies will someday be President. But it would be wrong to point to a particular kindergarten class of 25 kids and assert there's a 1 in a million chance one of them will become President, because of correlation with socioeconomic status of the community.

A quantitatively wrong guess is

• (for a young man) getting breast cancer sometime. NO:
Comment: My students are surprised to learn that men can get breast cancer; it's rare, but not so rare as they think, about 1 in 1,000 lifetime incidence, and 1 in 5,000 deaths. Chances for an individual vary with family history, but it's way more than 1 in a million.

• Being struck by lightning. NO
Comment: There isn't reliable data on being struck by lightning; if you don't seek medical attention you don't get into official statistics, and anyway can you tell the difference between lightning striking the tree you're under, or striking you? Here is data on U.S. deaths by lightning, which vary substantially from year to year but average around 60. Thus the population average is 1 in 5 million deaths per year, or about 1 in 70,000 lifetime. But neither figure is at all appropriate for a given individual. As I tell students, your grandmother is too sensible to be outdoors during a thunderstorm and around 30% of deaths are males aged 20-25. Chances for an individual vary hugely with their behavior, and there's no way to estimate an individual's chance to within a factor of 2.

Another familiar example:

• You being killed during a 150 mile auto trip in California. YES
Comment: The fatality rate in California is about 1 per 80 million vehicle miles; I scaled the numbers to account for multiple occupants and because you are a better driver than average.

Another iconic example is

• Casting the deciding vote in an election .....
This is an interesting classroom topic, because there are a variety of more or less sensible ways to analyze the question, and the answer depends on the circumstances. But the particular setting
• ....... in a California Statewide election that opinion polls say is too close to call
is a YES; here's the calculation that puts the chance at about 13/N where N is the number of votes cast, and there were about 12 million votes cast in California in the 2008 Presidenial election.

In general one should not use population averages as estimates for individuals, without reflection. For instance, this otherwise sensible page, as a final exercise, cites a list of "risks of 1 in a million [of dying]" including items such as

• smoking two cigarettes
• eating three hundred and fifty slices of stale bread (formaldehyde risk)
• drinking seventy pints of beer per year (alcohol cancer risk)
which are problematic for several other reasons too -- I would classify them as NO, if you were planning such an activity.

## Another treatment

See Understanding uncertainty: Small but lethal for a similar account, discussing small risks using the concept of micromort, a one-in-a-million chance of death.