Greenwood press

80
PROBABILITY

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of binomial situations can be estimated with normal distributions. (See Standard
Deviation.)
Probability arguments are common in court cases. DNA (deoxyribonucleic
acid) matching gives a probability that blood, semen, or hair found at a crime
scene matches the accused. The early cases of prosecution based on DNA match-
ing produced lengthy arguments about the accuracy of the techniques and the
computation of the probabilities. For example, some of the genetic markers that
are used in establishing probabilities occur in different proportions in different
racial groups. By 1996, recommendations from the National Research Council,
the National Institute of Justice, and other government and legal organizations
resulted in standardized laboratory techniques and computations of probabilities,
so DNA evidence is as well accepted as fingerprint matches. 
Probabilities have been used to determine whether juries were representative
in gender and racial composition to the communities they served. Lawyers for Al
Gore and George W. Bush used probability arguments before Florida courts to
persuade judges that their respective parties should prevail in the contested pres-
idential election of 2000. In the Microsoft antitrust case, the Department of
Justice presented probabilities that the Microsoft Corporation would force other
companies out of business. Courts have based financial awards to patients whose
cancer was misdiagnosed by doctors on computations of the patients’ reduced
probability of survival.
Probabilities can be computed from geometry formulas. Consider balls falling
uniformly on a square piece of cardboard 20 inches on a side that has a circular
hole 5 inches in diameter. The proportion of balls that fall through the hole is pro-
portional to the ratio of area of hole to area of the cardboard. This would be com-
puted using the formulas for area of circle and square: 
π(2.5)
2
20
2
≈ 0.049. A ball has 
about a 5 percent chance of falling through the hole rather than bouncing off the
cardboard. Winning carnival games is much more difficult than it appears!
online sources for further exploration
The geometry junkyard shows geometric probability problems

The birthday problem

Discrete probability

Diffusion
learn.htm>
Lottery odds calculations