文档介绍：Chapter 7—Probability Theory, Part 3 95
CHAPTER
Probability Theory,
7 Part 3
Variations of the Daughters Problem
A Note on Clarifying and Labeling Problems
Binomial Trials
Note to the Student of Analytical Probability Theory
The General Procedure
This chapter discusses problems whose appropriate concept
of a universe is not finite, whereas Chapter 8 discusses prob-
lems whose appropriate concept of a universe is finite.
How can a universe be infinite yet known? Consider, for ex-
ample, the possible flips with a given coin; the number is not
limited in any meaningful sense, yet we understand the prop-
erties of the coin and the probabilities of a head and a tail.
Example 7-1: The Birthday Problem, Illustrating the
Probability of Duplication in a Multi-e Sample
from an Infinite Universe (File “Birthday”)
As an indication of the power and simplicity of resampling
methods, consider this famous examination question used in
probability courses: What is the probability that two or more
people among a roomful of (say) twenty-five people will have
the same birthday? To obtain an answer we need simply ex-
amine the first twenty-five numbers from the random-number
table that fall between “001” and “365”(the number of days
in the year), record whether or not there is a duplication among
the twenty-five, and repeat the process often enough to ob-
tain a reasonably stable probability estimate.
Pose the question to a mathematical friend of yours, then watch
her or him sweat for a while, and pare your
answer to hers/his. I think you will find the correct answer
very surprising. It is not unheard of for people who know how
this problem works to take advantage of their knowledge by
making and winning big bets on it. (See how a bit of knowl-
edge of probability can immediately be profitable to you by
avoiding such unfortunate occurrences?)
96 Resampling: The New Statistics
More specifically, these steps answer the que