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
If you tossed the coins then the answer would be NO, unless I'm very confident
you lack the
ability to fool me ....
- 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.
One interesting example is
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!
- A major earthquake on the Hayward fault in the next 50 minutes. YES
Lest others become complacent, one can add e.g.
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.
- One of the next 25 babies born in the U.S. will become President. YES
A quantitatively wrong guess is
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.
- (for a young man) getting breast cancer sometime. NO:
What about our iconic case
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.
Being struck by lightning. NO
Another familiar example:
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.
You being killed during a 150 mile auto trip in California. YES
Another iconic example is
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
Casting the deciding vote in an election .....
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 a California Statewide election that opinion polls say is too
close to call
In general one should not use population averages as estimates for individuals,
this otherwise sensible page,
as a final exercise, cites a list of
"risks of 1 in a million [of dying]" including items such as
which are problematic for several other reasons too -- I would classify them as
NO, if you were planning such an activity.
- smoking two cigarettes
- eating three hundred and fifty slices of stale bread
- drinking seventy pints of beer per year (alcohol cancer risk)
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.