in the photograph, say, 10,000. The computer then
counts the number rof these points that hit the oil spill
shown in the photograph, yielding a proportion
that well approximates the fraction of the total area
covered by oil. For instance, if r= 7,568 points land in
locations corresponding to the spill, then the area of
the entire spill is very close to ×20 = 15.1 km2.
The greater the number of points selected, the more
accurate is the estimate.
In the same way, one can estimate the values of
complicated definite
INTEGRAL
s, such as ∫7
3cos2(x2+
sin(x))dx. For example, select a large number of
points at random from a box that contains the curve
in question and count the proportion of them that fall
below the given curve.
See also B
UFFON NEEDLE PROBLEM
;
PROBABILITY
.
Monty Hall problem Named after the host of a
popular American TV game show Let’s Make a Deal!,
the Monty Hall problem is a classic puzzler often used
to test initiates in the field of
PROBABILITY
theory. It
goes as follows:
On a game show, three closed doors stand
before you. The host informs you that a cash
prize lies behind one of the doors, with nothing
behind the other two. You select a door, but
before you open it, the host quickly opens one
of the remaining two doors to show you that
the prize is not there. He now gives you the
chance to change your mind and open instead
the third remaining door. The question is: what
should you do? Should you stay with your
original choice of door, or switch to the other
option? Is there any advantage to switching?
One’s typical first reaction to this puzzle is that
there is no advantage at all to switching. Since two
doors remain with only one containing a prize, the
chance of selecting the correct door, either by staying
with the chosen door or switching, is always 50 percent.
Surprisingly, this reasoning is not correct, for it makes
no use of the subtle information the host presents to
you, which you can actually use to your advantage.
Suppose, before you play the game, you decide that
you will stay with your choice. Then a win for you
relies on choosing the correct door initially, and there is
a 1-in-3 chance of this happening. If, on the other
hand, you play the game with the decision to switch,
then winning relies on choosing an incorrect door ini-
tially (this is where the host’s action comes to the fore),
and there is a 2-in-3 chance of this being the case. All
in all, we see in fact that switching doubles your
chances of winning! (This line of reasoning is made all
the more convincing if you imagine a game played with
100 closed doors, only one of which conceals a prize.
Choosing the correct door initially is very unlikely.
However, if the host reveals that 98 of the remaining
doors are empty, you will certainly decide that odds are
in your favor to switch.)
See also K
RUSKAL
’
S COUNT
;
TWO
-
CARD PUZZLE
.
Morley’s theorem In 1899 mathematician Frank
Morley (1860–1937) discovered that the intersections
of adjacent pairs of angle trisectors in any triangle
always form an equilateral triangle. Precisely stated,
given a triangle ABC, draw for each vertex a pair of line
segments dividing the angle at that vertex into thirds. If
the two line segments closest to side AB meet at point
D, the two segments closest to side BC at point E, and
the two segments closest to side AC at point F, then
DEF is guaranteed to be an equilateral triangle.
It is remarkable that this elegant fact of E
UCLIDEAN
GEOMETRY
was not discovered until so long after
Euclid’s time. The proof of this result, although a little
long and detailed, requires only very elementary geo-
metric techniques.
multiplication The process of finding the product
of two numbers is called multiplication. In elementary
arithmetic, multiplication can be defined as the pro-
cess of finding the total number of elements in a col-
lection of sets where each set in that collection has the
same number of elements. Thus, for example, the
accumulation of four sets each possessing three objects
gives a total of 12 objects. We write: 4 ×3 = 12. In this
context, multiplication can thus be regarded as a pro-
cess of repeated addition: 4 ×3 = 3 + 3 + 3 + 3 = 12. If
one arranges the 12 objects in a rectangular array of
four rows of three, then reading the arrangement as
three columns of four shows that 3 ×4 provides the
same answer as 4 ×3. In general, this reasoning
shows that products of counting numbers satisfy the
7,568
–––
10,000
r
–––
10,000
multiplication 343