18 1. THE ORIGIN AND PREHISTORY OF MATHEMATICSare equally likely. Hence the probability of this new event F is 1/2. Thus, even
though the mathematics of conditional probability is quite simple, it can be a subtle
problem to describe just what event has occurred. Conclusion: To reason correctly
in cases of conditional probability, one must be very clear in describing the event
that has occurred.
1.21. Reinforcing the conclusion of Problem 1.20, exhibit the fallacy in the follow-
ing "proof" that lotteries are all dishonest.
Proof. The probability of winning a lottery is less than one chance in 1,000,000
( = 10-6). Since all lottery drawings are independent of one another, the probability
of winning a lottery five times is less than (10~^6 )^5 = 10~^30. But this probability
is far smaller than the probability of any conceivable event. Any scientist would
disbelieve a report that such an event had actually been observed to happen. Since
the lottery has been won five times in the past year, it must be that winning it is
not a random event; that is, the lottery is fixed.
What is the event that has to occur here? Is it "Person A (specified in advance)
wins the lottery," or is it "At least one person in this population (of 30 million
people) wins the lottery"? What is the difference between those two probabilities?
(The same fallacy occurs in the probabilistic arguments purporting to prove that
evolution cannot occur, based on the rarity of mutations.)
1.22. The relation between mathematical creativity and musical creativity, and the
mathematical aspects of music itself are a fascinating and well-studied topic. Con-
sider just the following problem, based on the standard tuning of a piano keyboard.
According to that tuning, the frequency of the major fifth in each scale should be
3/2 of the frequency of the base tone, while the frequency of the octave should
be twice the base frequency. Since there are 12 half-tones in each octave, starting
at the lowest A on the piano and ascending in steps of a major fifth, twelve steps
will bring you to the highest A on the piano. If all these fifths are tuned properly,
that highest A should have a frequency of (|)^2 times the frequency of the lowest
A. On the other hand, that highest A is seven octaves above the lowest, so that,
if all the octaves are tuned properly, the frequency should be 2^7 times as high.
The difference between these two frequency ratios, 7153/4096 « 1.746 is called the
Pythagorean comma. (The Greek word komma means a break or cutoff.) What
is the significance of this discrepancy for music? Could you hear the difference
between a piano tuned so that all these fifths are exactly right and a piano tuned
so that all the octaves are exactly right? (The ratio of the discrepancy between the
two ratios to either ratio is about 0.01%.)
1.23. What meaning can you make of the statement attributed to the French poet
Sully (Rene Frangois Armand) Prudhomme (1839-1907), "Music is the pleasure the
soul experiences from counting without realizing it is counting"?