## by Frederick
Mosteller

### A look at the essence of probability, at all levels

### July 2001

If Mosteller hadn't included the solutions, this would have been a short book
indeed -- 56 problems simply stated in 14 pages. You'll soon find, however, that
some problems, which are the shortest to set up, take a great deal of brainpower.
It starts innocently enough - some simple-sounding problems on socks in drawers,
flipping coins, and rolling dice. Soon enough, you end up with paper black with
numbers and pictures of a flipping coin (how thick does a coin need to be so that
it lands on its =side= with probability 1/3?) If you get drawn in deep (as I
did), you may even wonder what probability really means.

Some of the problems are classic, such as the problem of how many people would it
take for the probability that at least two of them have the same birthday is
greater than a half (I'll give this answer away: 23. But do you know why?) One
of the dice problems actually recalls the history of the development of
probability as a separate mathematical field -- problem #19, involving dice bets
that Samuel Pepys asked Isaac Newton to figure out. Some of the problems are
simply openers for entire vistas in probability - avoid problems #51 and #52 if
you wish to not become enmeshed in concerns of random walks (remember that one of
Einstein's earliest papers was on Brownian motion - a molecular random walk.) I
used problem #25, which deal with "random chords on a circle", to explore this
classic probability paradox - I've ended up with three different figures, all of
which seem plausible! It gets deep to what one means by "random chord".

This book, though so thin, is inexhaustible in spawning disturbing questions
about probability; even more useful is that there are questions for people at
=any= level of knowledge of probability. Those who wish to think about
"counting" problems (like those involving rolling dice, or pulling balls out of
urns) will find those here. Those who have an interest in continuous probability
will find problems which will interest them. And those old probability pros who
ponder the essence of chance will find meat for some productive chewing.

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Mary Pat Campbell, last updated July
2001