SPREADING RUMORS
- Problem. Your best friend tells you a secret and
makes you promise not to tell anyone. Although you are just
dying to tell someone, you don't really want your friend's
secret to be widely known. Is there much harm in telling
just two people?
- Intuition. When spreading rumors, our sense is
that there is little harm in telling just one or two
people. We make them promise not to tell anyone else (in
the same way that we promised), and figure it will stop
there. Or, if they do tell someone, then the rumor won't
spread very far. As we will see below, this is an example
of where our intuition is quite poor: rumors spread very
fast.
- Math. You tell your best friend's secret to two
people. Now, 4 people know the secret: you, your best
friend, and the two people you just told. Let's assume
that the two people to whom you told the secret each tell
two people, for a total of four new people. Now, 8 people
know the secret. If each of the last four people to
receive the secret tell two people, eight new people know
the secret. After four passes of the secret, a total of 16
people know the secret.
This process continues: the group of people who just
learned the secret, each tell two more people. After just
ten passes, 1,024 people will know the secret, and after
20 passes, 1,048,576 people will know the secret. After 28
passes, 268,425,456 people know, which is roughly the
population of the United States. And after 33 passes, more
than 8.5 billion people will know, which is more than the
population of the entire planet.
The number of people who know a secret doubles after each
pass (2, 4, 8, 16, 32, ...). That is, after N passes, 2N
people know the secret. This is known as exponential
growth.
If we assume that each person will act in a similar manner
to yourself, we see that the rumor will spread to a large
number of people very quickly.
- Summary. When holding a secret we believe that
little harm is done by telling just a few people, since the
number of people that know the secret has only increased by
a small number. The problem, of course, is that everyone
thinks like this, and everyone tells a few people. The
number of people who learn of the secret, therefore, grows
very quickly, at a exponential rate. Here are a few more
surprising examples of the impact of exponential growth.
- Take a simple piece of notebook paper. Fold it in
half. Fold it in half again, and again. How thick will
the paper be after a total of 50 folds? A single sheet
of paper is 0.1 mm thick. Folded once, the paper is
0.2 mm thick; folded twice it is 0.4 mm thick; folded
three times it is 0.8 mm thick. The thickness is
doubling each time (exponential growth). After ten
folds, the paper is 102.4 mm thick (10.24 cm = 4.03
in). After twenty folds, the paper is 104,857.6 mm
thick (approximately 105 m or 344 ft -- approximately
the height of a 30-story building!). After 50 folds,
the paper is 112,589,990,684,262.4 mm thick
(approximately 112 billion km or 70 billion miles,
almost the distance to the sun.
- Once upon a time there was a king who loved to
play chess. One day the king proposed that if anyone
in his kingdom could beat him at chess, the king
would grant them any reasonable wish. A poor farmer
stepped forward to meet the king's challenge. Much
to the king's surprise, the farmer beat the king
quickly and with seeming ease. True to his word, the
king agreed to grant the farmer's wish. Wanting to
wish for something that seemed reasonable, the
farmer suggested the following: "I propose that you
place on the first square of the chess board one
penny, and on the second square, two pennies, on the
third, four pennies, and so forth, until the last
square is reached." After a moments thought, the
king granted the request, as it seemed that the
farmer had, in fact, asked for very little money. As
the king began to place the money on the chess
board, however, he soon realized his terrible
mistake. Let's see why.
The chess board consists of 8 x 8 = 64 squares. On the first row of the board, the king placed:
0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28
for a total of $2.55. Notice that the
amount of money is doubling between squares
(exponential growth). On the second row, the king
placed:
2.56, 5.12, 10.24, 20.48, 40.96, 81.92, 163.85, 327.68
Combined with the first row, there was a
total of $655.35 on the board. On the third row, the
king placed:
655, 1310, 2621, 5242, 10485, 20971, 41943, 83886
Combined with the first two rows, there was a total
of $167,772.16 – and there were still five rows to
complete! Continuing along like this, the completed
board would have contained
$368,934,881,474,191,032.32. The king, of course,
ran out of money well before this and was unable to
keep his promise.
RISK: IS IT WORTH IT?
- Problem. You are thinking of cheating on your
math test. If you cheat, and don't get caught, you will
get a better grade. If you cheat, and get caught, you will
fail the exam and get in trouble. Putting aside the moral
and ethical issues, how do you decide if you should cheat
or not?
- Intuition. Your decision is based on what you
expect to gain by cheating balanced against the chance of
getting caught and the penalty for getting caught. Let's
say that if you don't cheat, you expect to score 60/100 on
your exam. If you do cheat you expect to score 100/100. We
need to factor in one more thing. If you cheat and get
caught, you will score 0/100 (let's ignore, for now, the
other penalties associated with getting caught
cheating). Naturally, you will decide to cheat if doing so
will increase your exam score. But since you don't know
beforehand if you will get caught or not, you don't know
if cheating will yield a score of 100 or 0.
- Math. Let's say that the chance of getting
caught is 50%. This means that the probability of getting
caught is 50/100 or 1/2. The way to think about
probabilities is to imagine taking, and cheating on, a
large number of exams. After taking all of these exams,
you can expect to get caught 1/2 of the time. On half of
the exams you will score 100 and on the other half you
will score 0. So your average score is 50 (100 x 1/2 + 0 x
1/2). This means that on any single exam, your expected
score, if you cheat, is 50. This is the score that we
compare against the score you receive for not cheating,
60. In this case, cheating won't pay off.
Perhaps it seems strange to say that you expect to score
50 on the exam, since you will never actually receive a
score of 50 (you will score 100 or 0). This expected score
arises because of the uncertainty in getting caught - it
allows us to combine the possible scores with their
probabilities so that we have a single number upon which
we can make a decision.
Let's now say that the chance of getting caught goes down
to 25%. In this case you will get caught 25/100 = 1/4
times, and not get caught 75% = 75/100 = 3/4 times. Your
expected score now is 75 (100 x 3/4 + 0 x 1/4). In this
case, cheating does pay off – you can expect to gain an
extra 15 points over not cheating.
Now, let's factor in the other penalties for cheating. You
will get in trouble with your teacher, principal and your
parents. You might get detention, suspended, lose TV/video
game privileges, etc. These should be factored into your
decision. Let's say that the punishment is the equivalent
of 200 exam points. Now, the score you receive for
cheating and getting caught is -200 (instead of just
0). Your expected score now becomes 25 (100 x 3/4 - 200 x
1/4). In this case, cheating does not pay off because it
is less than the score of 60 that you would receive by not
cheating.
- Summary. When faced with a decision where the
outcome of one or more of your choices is uncertain, you
can make a mathematically sound decision by combining the
cost and probability associated with each option. The
expected cost of any option is simply the probability
times the cost. This strategy applies to many real-world
situations:
- You can buy your favorite candy bar (A) for $2, or
two of a new candy bar (B) for $1 each. Which should
you buy? Let's assign a score of 100 to the pleasure
of eating candy bar A. You have never tried candy bar
B before, so you don't know if it will bring you more
pleasure or not. Let's say that the probability of you
liking candy bar B as much as A is 25% = 25/100 =
1/4. Then, the expected pleasure of candy bar B is 25
(100 x 1/4), and the expected pleasure of both of them
is 50 (100 x 1/4 + 100 x 1/4). In this case, you
should buy candy bar A – quality wins over
quantity. If however, the price of candy bar B was
just $0.25, then you could buy eight of them for $2,
and your expected pleasure would be 200 (8 x 100 x
1/4), and you should try the new candy bar – quantity
wins over quality.
- A lottery ticket costs $1 and the jackpot is
$1,000,000. Should you buy the ticket? We need to
compare the cost of the ticket ($1) with the expected
payoff of buying the ticket. Let's say that the chance
of winning is 1/50,000,000 (this is roughly the chance
of winning the New York State lottery). The expected
payoff is 2 cents (1,000,000 x 1/50,000,000). In this
case, you should not buy a ticket.
- Your football team just scored a touchdown. You can
kick a field-goal to add 1 point to your score, or try
for 2 more points by running/passing the ball into the
end-zone. Which should you do? Let's say that the
probability of making the field-goal is 90% = 90/100 =
9/10, and that the probability of making the 2-point
conversion is 25% = 25/100 = 1/4. Then, the expected
payoff of the field goal is 0.9 points (1 x 9/10) and
the expected payoff of the 2-point conversion is 0.5
(2 x 1/4). In this case, you should kick the
field-goal (assuming that you are not down by 2 points
with 10 seconds left on the clock).
ARE YOU SURE ABOUT THAT DOC?
- Problem. During a routine check-up, your doctor
runs a test that reveals that you have chicken pox. You
know that the test might be wrong, so how do you determine
the chance that you are actually sick?
- Intuition. The doctor tells you that the test
will report that you have chicken pox when you do not, one
out of every 1,000 times. In other words, the test is
99.9% accurate, suggesting that there is a very good
chance that you have chicken pox.
The accuracy of the test, however, is not the
only thing that we need to consider. We also need to
consider the chance that you would be sick, regardless of
the test. For example, a young boy tests positive on a
pregnancy test that is wrong 1 out of 1,000 times. Surely
we wouldn't say that the chance that he is pregnant is
99.9%. We need to consider the fact that it is impossible
for him to be pregnant (test or no test).
- Math. The probability that you are sick given a
positive test result is the probability that the test
result is positive given that you are sick, times the
probability that you are sick (this is known as Bayes'
rule, after the mathematician Thomas Bayes, 1702-1761). We
can express this more formally as:
P(S|T) = P(T|S) P(S)
where P() denotes the probability of an event, S=you are
sick, T=positive test result and "|" should be read as
"given" (e.g., P(S|T) is the probability that you are sick
given a positive test result).
For example, the probability that the chicken pox test is positive given that you have chicken pox is:
P(T|S) = 0.999
this is the known error of the test. If your younger
brother came down with the chicken pox, and you have never
before had chicken pox, then the chance that you have
chicken pox is pretty high, let's say 90%:
P(S) = 0.90
The overall probability that you have chicken pox is then:
0.999 x 0.90 = 0.8991
or just under 90%. If, however, you had chicken pox two
years ago, and you have not come into contact with anyone
with chicken pox, then the probability that you have
chicken pox is low, let's say 20%:
P(S) = 0.20
The overall probability that you have chicken pox is then:
0.999 x 0.20 = 0.1998
or just under 20%, much smaller than the 99.9%
based only on the test.
- Summary. When evaluating the results of a
medical test, you should take into consideration your
inherent risk of having contracted whatever it is that you
may have tested positive for. A surprising number of
doctors don't understand this. So the next time the doctor
returns to you a test result, ask them if their diagnosis
has taken into consideration the full Bayesian likelihood.
ART: PACKING THREE DIMENSIONS INTO TWO
- Problem. We live in a three-dimensional spatial
world. Three dimensions because you can move left-right,
up-down, and forward-backward. An artist's canvas is
two-dimensional: the paint brush cannot move in the
forward-backward direction when confined to the
canvas. How then does an artist create a sense of
three-dimensional space on a two-dimensional
canvas?
- Intuition. We perceive the buildings and road in
the painting shown below as receding away from us, even
though they are all simply flat objects on our computer
screen. Our intuition probably tells us that the reason
for this three-dimensional perception is that the
buildings on the right (further from us, in depth) are
painted smaller than those on the left (closer to us, in
depth). Similarly, the road narrows as it recedes away
from us.
In as early as the 1400s, artists were conveying a sense
of three-dimensional structure on a flat two-dimensional
canvas. As we will see below, the key insight that led to
this was the understanding of some simple
geometry.
- Math. Shown below is a diagram of how an artist
might map a three-dimensional shape onto a two-dimensional
canvas. In the lower part of the diagram, the object (red
rectangle) is closer to the artist. In this case, the
"image" of the object on the canvas is larger than when
the object is further. That is, in order to convey a sense
of three-dimensional structure, nearby objects should be
drawn larger than distant objects. The precise difference
in size is determined by geometry.
Notice that in the above diagram we have a pair of similar
triangles (gray). The ratio of the sides of similar
triangles are equal, meaning that u/f = X/Z or u = fX/Z
. That is, the size of the object on the canvas, u, is
inversely proportional to its distance Z. This equation,
known as perspective projection, specifies how large an
object, that is a given distance from you, should be
drawn.
Shown below is an example of how to sketch a box under
perspective projection to give the correct sense of
three-dimensional structure.
- Draw a horizontal line (red). This is the horizon,
and the ends of the line are the vanishing
points. Draw a single vertical line (black). This is
the front edge of the box. Connect the ends of this
line to the vanishing points.
- Draw a vertical line (black) inside of the triangle
formed by the lines extending from the left-most
vanishing point. This is another edge of the
box. Connect the ends of this vertical line to the
right-most vanishing point (gray).
- Repeat the above step for the other side of the
box. iv.Erase the horizon and guidelines, leaving
behind a sketch of a three-dimensional box.
- Summary. An artist's canvas is
two-dimensional. And yet, artists are capable of giving us
a sense of the three-dimensional space of the scene being
painted. This is accomplished by painting nearby objects
larger than more distant objects. The rules of geometry
guide an artist in achieving the correct proportions in
their paintings.
PRIME NUMBERS AND THE NATIONAL SECURITY AGENCY
- Problem. The National Security Agency (NSA)
wants to be able to send email in such a way that if
somebody intercepts the message, they will not be able to
read the message. And at the same time, the person for
whom the message was intended should be able to read the
message.
- Intuition. In order to send a message that
nobody but the intended party can read, the sender and
receiver need to agree upon some kind of code so that a
message can be scrambled and un-scrambled. The message
"meet in prague on oct eleven at ten", for example, could
be scrambled to read "teem ni eragup no tco nlevee ta
net". Here, the first and last letters were switched. If
this scrambled message were intercepted, its contents
would not be immediately obvious. But, it probably
wouldn't take long for somebody to break the code and
decipher the message.
The NSA, therefore, needs a code that makes it is easy for
the sender to encode a message, and for the receiver to
decode the message, but that is very difficult for anybody
else to decode.
- Math. A commonly used code for sending secure
messages is known as RSA encryption. There are three basic
steps:
- Generate "keys"
- - select two large prime numbers, p and q
- - compute the product of the two primes, n = pq
- - compute the following product, m = (p-1)(q-1)
- - choose a number e so that the greatest common divisor of e and m is 1
- - compute d so that 1 < d < e and ed mod m = 1 (ed mod m is the remainder of dividing ed by m as many times as evenly possible)
- Scramble message (encryption)
- - convert each letter of the alphabet to a number (a=1, b=2, c=3, ..., z=26, A=27, ..., Z=52)
- - replace each letter of the message, x, with y = x^e mod n (^ means raise x to the power e)
- - send the scrambled message, y
- Unscramble message (decryption)
- - replace each letter of the received message, y with y^d mod n
- - the result will be the original text message
- Summary. Mathematicians and computer scientists
have figured out a clever code for sending secrets that,
for now at least, is very very difficult to break. The
code relies on the simple numerical property that it takes
computers a very very long time to factor large numbers
into a product of prime numbers.
The NSA (and others) is, by the way, trying to develop a
special type of computer known as a quantum computer that,
if it can be built, will be able to break this
code.
MAKE YOUR PARENTS PROMISE TO BUY YOU A MOTORCYCLE
- Problem. You want a motorcycle, but your parents
won't let you buy one.
- Intuition. Your parents are unlikely to simply
agree to buy you a motorcycle. And it is unlikely that you
can convince them to do so by cleaning your room, taking
out the garbage, being nice to your little brother,
etc. You need to get them to agree to buy you a
motorcycle, without them knowing that they are doing
so.
- Math. If your parents are true to their word,
all it will take for them to agree to buy you a motorcycle
is to have the following conversation with them.
you "Will you promise to hug me if I make a true
statement and not hug me if I make a false statement?"
parents "Uh, sure."
you "You will neither give me a hug nor will you buy me a motorcycle."
That's it. Now, explain to your parents their options:
- Your parents hug you, but they don't buy you a
motorcycle. If your parents hug you, then your
statement ("no hug, no motorcycle") will be false. But
your parents agreed not to hug you if you made a false
statement – so this option is no good.
- Your parents don't hug you, and they don't buy
you a motorcycle. In this case, your statement ("no
hug, no motorcycle") will be true. Your parents
agreed to hug you if you made a true statement. But
we just agreed above that they can't hug you – so
this option is no good.
- Your parents hug you, and they buy you a
motorcycle. In this case, your statement ("no hug,
no motorcycle") is false. But your parents agreed
not to hug you if you made a false statement – so
this option is no good.
- And lastly, your parents don't hug you, but they
buy you a motorcycle. In this case, your statement
("no hug, no motorcycle") is false. Since your
parents agreed not to hug you if you made a false
statement, there is no contradiction here.
Option (4) is the only option that is consistent with your
parent's promise, so they must buy you a motorcycle if
they are to keep their promise to you.
The above is an example of the application of a branch of
mathematics known as logic. We can simplify the above
statements with a few mathematical symbols:
- H = your parents hug you
- ~H = your parents don't hug you
- M = your parents buy you a motorcycle
- ~M = your parents don't buy you a motorcycle
- S = (~H and ~M) – this is your statement
- T = a true statement
- F = a false statement
Your parents promised that (if S=T then H) and (if S=F
then ~H). We can now write the four options as follows
(the symbols in red show the contradiction):
- (H and ~M) implies S=F implies ~H
- (~H and ~M) implies S=T implies H
- (H and M) implies S=F implies ~H
- (~H and M) implies S=F implies ~H
Only the last option has no contradiction.
- Summary. The above "trick" worked because we
created a set of logical statements that led to
contradictions in all but the desired conclusion. The
logic of these statements is made particularly obvious
when the statements are replaced with concise symbols.
By the way, I doubt that this will actually convince your
parents to buy you a motorcycle. But, maybe your parents
will be sufficiently impressed with how clever you are,
that you can convince them to buy you a video game as a
consolation for them breaking their promise.