Adrian Roberts writes:
>According to the author the expert witness at the recent trial, where a
>solicitor was convicted of smothering her 2 infant children, used
>probability theory incorrectly in advising the court. The author explains
>the error made in terms of arrows in targets drawn before versus after the
>arrows are shot. I can appreciate that there is something quite important
>here that I (and lots of other ordinary medics who glibly bandy
>figures/statistics around) really need to understand. So far I have
>failed to get my brain round it.
>
>Has anyone a *simple* explanation of the problem for the statistically
>ungifted. Perhaps something using throws of dice or tossing of coins?
In general, statistics are good at characterizing the behavior of groups,
but do not do so well at characterizing the behavior of individuals. For
groups, you can usually get some help from the Central Limit Theorem and/or
the Law of Large Numbers. You don't get this type of help for a sample of
size 1.
Furthermore, a lot of these "courtroom probabilities" are problematic,
because you need to carefully examine the frame of reference.
A friend of mine found a dollar bill that had an interesting pattern in the
serial number. It was something like 55773366. He asked what was the
probability of this happening. But the probability depends on your frame of
reference. Are you referring to the particular serial number, or any serial
number of the form aabbccdd? Is order important, or would a pattern like
abcdabcd also count? Are you looking at the probability for a single dollar
bill or the probability of seeing that in all of the dollar bills that you
come across in a day's time? a week's time? etc.
One analogy I like to think of is that of the lottery. Any individual
person's chances of winning the lottery are very small, but by golly someone
does seem to win just about every week. It depends on your frame of
reference (what is the chance I will win versus what is the chance that
someone will win).
Another problem is lack of independence, though this is less of an issue.
The probability of encountering a red Corvette is not equal to the product
of the probability of red and the probability of Corvette, because red is
more popular among Corvette owners than other car owners.
Many years ago, I heard an excellent talk by Stephen Fienberg (a famous
statistician) about the difficulty with using probability arguments in a
courtroom. One piece of advice he gave is to be very skeptical of these "one
in a million" probabilities, as they are usually based on questionable
assumptions.
This is an important issue, because it affects other things, such as report
cards for hospitals. What's the probability that a hospital is two standard
deviations below the mean. The probability is quite high actually if you
grade enough hospitals.
I hope this helps.
Steve Simon, [log in to unmask], Standard Disclaimer.
STATS - Steve's Attempt to Teach Statistics: http://www.cmh.edu/stats
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