This article may be useful as supplementary reading for
mathematics classes, based on the DSE syllabus.
根據數學科文憑試課程綱要，本文或可作為
有用的補充讀物。
The
f i r s t a cco u n t s
of coin-fl ipping
d a t e a l l t h e
way back to the
Romans, where the
c h a n ce o u t come
of either heads or tails
was be l i eved to be a
man i fes tat i on of d i v i ne
will. While nowhere near as
ser ious, coin-flipping today
is still considered as an unbiased way to reach a
decision or settle a dispute. In certain sports, it is
commonly used to decide which side each team
plays on, or to select the winner in the case of a
tie. But just how random is the coin flip? A former
professional magician turned statistician, Persi
Diaconis, was interested in exploring this question.
Diaconis and his colleagues carried out simple
experiments which involved flipping a coin with
a ribbon attached. By unwinding the ribbon from
the flipped coin, the number of times the coin had
rotated was determined. To eliminate undesirable
variations in the coin toss, the initial conditions of
the coin toss must be consistent. Thus, a coin-tossing
machine was used, where the coin was placed
on a spring which was released by a ratchet.
Additionally, a high-speed slow motion camera
was employed to capture 100 frames of 2D images
for each coin toss. The 2D images enabled them
to measure the orientation of the coin mid-flip with
angled precision [1].
They found that when the initial conditions were
the same, the coin flip would produce the same
result, implying that the unpredictability of the coin
toss is most likely caused by human inconsistencies.
Additionally, when the coin is tossed by hand, there
is a slight bias of a 51% chance that the coin lands
o n
t h e
s a m e
f a c e a s
i t was tos sed
(tos sed as heads ,
results in heads). When a
coin is spun on a surface, it is biased to land with its
heavier side down [2]. With these biases in place,
some magicians and gamblers are able to perform
“controlled coin flips” - the coin is hit exactly in
the centre so that its angular momentum vector
lies perpendicular to the coin, causing it to go up
without turn
lands on the face that it started with pre-toss. They
may also use a coin with slightly shaved edges to
manipulate the coin to always land on a certain
face when spun [1].
In fact, even ‘random’ number generators
in our computers are not entirely random. Steve
Ward, a Computer Science and Engineer ing
professor at MIT, states that by following rules and
relying on algorithms, computers are specifically
programmed to be deterministic, which means that
they give the same answer to the same question
every time [3]. The computer generators are known
as pseudorandom number generators (PRNGs),
which are algorithms that take in a seed number to
generate a sequence of numbers that approximate
random numbe r s . The refo re, t he ‘ random’
sequence can actually be reproduced if the seed
value is known. Truly stochastic occurrences are
physical phenomena such as radioactive decay
or cosmic background radiation, which can be
measured over short timescales.
However, several caveats were not taken into
account in Diaconis et al.’s experiments. Human-
generated coin flips are subject to variations
including the height and speed of the toss, and the
By Jacqueline Aw
歐婷梅
Heads
or
Tails?
