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PROBABILITY
Probability
Probability is a measure of how likely something
is to happen. If you flip a coin, the probability of
the coin landing heads is 50%, meaning that you
expect it to land heads 50 times out of every
100 flips. If you roll a die, the probability of the
die landing on 4 is 1/6 (because a die has 6
faces), meaning that on average, you would roll
a 4 once every 6 rolls.
Conditional Probability
Sometimes the probability of an event is
increased or decreased by other events. The
probability that there is no final for this class is
very low. But IF I die, the probability is much
higher. And IF Lingnan closes, the probability is
much much higher. We say that the probability
of P IF Q is the probability of P conditional on Q.
Representing Probabilities
We can represent probabilities using the symbol
P(-). [This is a little confusing, because before we
were using P to represent sentences.] For
example:
P(H) = 50%
This might mean “the probability of the coin
landing heads is 50%.”
So for example, before we learned that the
probability of A happening is always greater
than or equal to the probability of A and B
happening. We can represent this truth as
follows:
P(A) ≥ P(A & B)
Conditional Probabilities
We also have a way of representing conditional
probabilities: P(A/B) means “the probability of A
conditional on B” or “the probability that A will
happen IF B happens.”
P(~F/C) > P(~F)
The probability that there will be no final
conditional on Lingnan closing is greater than
the probability that there will be no final.
Review: Which of the following two
statements is true?
1. P(Fido is an animal/
Fido is a dog) = 100%.
2. P(Fido is a dog/ Fido is
an animal) = 100%.
Experiments
Scientific Method
Science proceeds by the hypothetico-deductive
method, which consists of four steps:
1. Formulate a hypothesis
2. Generate testable predictions
3. Gather data
4. Check predictions against observations
Experiments
Today we’re going to talk about experiments and
good experimental design.
How do we design experiments that can test our
hypotheses? Experiments that can generate
data that are relevant to our predictions?
Causation
Much of science is concerned with discovering
the causal structure of the world.
We want to understand what causes what so we
can predict, explain, and control the events
around us.
Prediction
For example, if we know
that rain is caused by cool,
dry air meeting warm, wet
air then we can predict
when and where it will
rain, by tracking air
currents, temperature,
and moisture.
Prediction
This is important because
rain affects our ability to
engage in everyday
activities, like traveling or
exercising.
Knowledge of causation
lets us make predictions,
which helps us make plans
Explanation
One way to explain
something is to determine
what causes it.
For example, if you find
out that a certain virus
causes a disease among
bears, then you have
explained why the animals
are getting sick.
Explanation
This is important because
once you know an
explanation for a disease
(what causes it), you can
begin treating it– for
example, with antiviral
drugs.
Control
Finally, if we know what
causes some effect, then
we can control nature to
our advantage.
For example, if you don’t
know what causes
diamonds, you have to
look through mines to find
some.
Control
But when we know that
diamonds are caused by
carbon under high
pressure, high
temperature conditions,
we can simply re-create
those conditions to grow
as many diamonds as we
want.
CAUSATION VS. CORRELATION
Independence
In statistics, we say that two variables are
independent when the value of one variable is
completely unrelated to the other:
P(A/ B) = P(A) and P(B/ A) = P(B)
B happening does not make A any more likely to
happen. (If that’s true, so is the reverse.)
Example
For example, recall one of our non-random
sequences of coin flips:
XOXXOXOXOOXXOXOOXOXO
How did we know that this sequence was nonrandom? Because whether the coin lands X or O
is not independent of the other tosses.
Example
For example, recall one of our non-random
sequences of coin flips:
XOXXOXOXOOXXOXOOXOXO
P(X/ O) = 7/9, P(X) = 10/20
P(O/ X) = 8/10, P(O) = 10/20
Correlation
Two variables A, B that are not independent are
said to be correlated.
A and B are positively correlated when P(A/ B) >
P(A). If B happens, A is more likely to happen.
A and B are negatively correlated when P(A/ B) <
P(A). If B happens, A is less likely to happen.
Correlation
Other relationships between variables are often
called correlation as well.
A and B are positively correlated when increases
in A correspond to increases in B.
A and B are negatively correlated when
increases in A correspond to decreases in B.
Positive Correlation Example
For example, demand and
price are positively
correlated.
If demand increases for a
certain product, then the
price of that product
increases. If demand
decreases, price
decreases.
$250,000 for 1 Rhino Horn
A greatly increased
demand for rhino horn in
traditional Chinese
medicine has led to a
tremendous price increase
for the horns.
They are worth so much
now that all 5 species of
rhino are close to
extinction.
Negative Correlation Example
On the other hand, supply
and price are negatively
correlated.
If supply increases for a
certain product, then the
price of that product
decreases. If supply
decreases, price increases.
Pork Prices Predicted to Soar
So recently, higher corn
prices have made pigfarming less profitable,
leading to a decreased
supply of pigs.
Experts are predicting that
there will be an increase
in pork prices next year.
Causation and Correlation
One thing that can lead two variables A and B to
be correlated is when A causes B.
For example, if having a cold causes a runny
nose, then having a cold is correlated with
having a runny nose:
P(cold/ runny nose) > P(cold)
Causation and Correlation
Similarly, the number of cars on the road is
correlated with the number of accidents: if
there is an increase in the number of people
driving, there will be an increase in the number
of car accidents.
This is because a larger number of cars causes a
larger number of accidents.
Causation ≠ Correlation
But causation does not imply correlation. If A
and B are correlated there are several
possibilities:
•
•
•
•
A causes B
B causes A
C causes A and C causes B
A and B are only accidentally correlated
B causes A
Whenever there are lots
of police at a location, the
chance that there is a
criminal there goes up.
So do police cause crime?
No, exactly the opposite:
crime causes the police to
show up!
B causes A
Here’s a somewhat more realistic example. It
has been observed that democracies tend to get
in fewer wars than non-democratic countries.
A plausible inference would be that the negative
correlation between democracy and war is due
to the fact that democracy causes peace.
B causes A
But there’s another explanation, and some
studies have suggested that it’s the right one.
Frequent wars cause a country to not be
democratic. Countries that get in a lot of wars
don’t have the stability that’s necessary for
democracy to flourish.
Common Cause
Sometimes A and B are correlated, not because
A causes B and B causes A, but instead because
a third variable C, the common cause, causes
both A and B.
Porn and Rape
A study of U.S. prison
inmates found that
prisoners who had been
exposed to pornography
earlier in life were less
likely to be in prison for
rape, compared with
those exposed to porn
later in life.
Porn and Rape
Does this mean that exposure to porn early in
life prevents men from becoming rapists?
Should you give your children porn?
No. Inmates who had been exposed to porn
later were more likely to have had a religious
fundamentalist upbringing.
Porn and Rape
And a religious fundamentalist upbringing was
correlated with higher rates of sexual deviancy
(and rape).
Fundamentalist upbringing caused both late
exposure to porn and higher chances of sexual
crimes.
Coincidence
The “Texas Sharp Shooter”
Suppose I stand in front of a barn. I have a
machine gun with me, and I am blindfolded. I
shoot wildly at the barn for several minutes.
Afterward, I walk up to the barn. I find a spot
where three bullets are very close together, and
I paint a target around them. “Look!” I say, “at
what an excellent marksman I am!”
Rare Things are Frequent
Rare coincidences are bound to happen
sometimes. How likely is it that someone will
both win the lottery and get struck by lightning?
Well, there is 1 lottery every week, 50 every
year. In a span of 30 years, 1500 people will win
the lottery.
Getting Struck by Lightning
There is a 1 in 1 million chance of getting struck
by lightning in any given year. Let’s suppose
each lottery winner on average lives 30 years
after winning. That’s 30 distinct 1 in 1 million
chances of getting struck, or a 30 in 1 million
chance of getting struck in 30 years.
P(struck) = 1 – P(not struck) = 1 – .999999^30
Winners Getting Struck by Lightning
So what’s the probability that any of the 1,500
winners will get struck?
P(some winner is struck) = 1 – P(no winner is
struck) = 1 – .99997^1500 = 1 – .955997 = .044 =
4.4%.
That’s higher than the probability that a coin will
land heads 5 times in a row.
Lucia de Berk
In 2006, Lucia de Berk, a
nurse at a hospital in the
Netherlands was
convicted of killing 7
children.
There was no evidence
against her except for the
fact that she was in the
room during or before
each of the deaths.
Correlation
Prosecutors reasoned that there was a
correlation: Lucia de Berk in the room & death.
It couldn’t be that the deaths caused her to be
in the room.
It couldn’t be that some common cause C both
caused her to be in the room and the deaths.
So the only other option was that she caused
the deaths.
Coincidence
But there was a third option: coincidence.
How many hospitals are there in all the world?
How many nurses work at each of those
hospitals? What are the chances that, just by
accident, in one of those hospitals one of those
nurses just happened to be present for 7
deaths?
Rare Things are Frequent
Richard Gill, Professor of Mathematical Statistics
at the University of Leiden, worked hard to
overturn the case. He estimated that the chance
that this was an accident was 1 in 9.
This doesn’t prove that she’s innocent (or
guilty). But things that have a 1 in 9 chance of
happening happen all the time!
EXPERIMENTAL DESIGN
Types of Scientific Studies
There are two basic types of scientific studies
(the stuff that gets published in scientific
journals and reported in the “science” section of
the newspaper):
• Observational studies
• Controlled experiments
Observational Studies
An observational study looks at data in order to
determine whether two variables are correlated.
Observational Studies
For example, an
observational study might
ask women to record how
much wine they drink, and
also to report if they
develop breast cancer.
After many years, a
correlation may be found
between wine
consumption and cancer.
Importantly, observational studies can only
show whether two variables A and B are
correlated. They cannot show whether A causes
B, or B causes A, or some third cause causes
both, or if the correlation is accidental.
Controlled Experiments
The first recorded
controlled experiment
occurs in the Book of
Daniel, part of the Jewish
Torah and the Christian
Bible.
Daniel’s Experiment
Daniel wanted to discover
which of two diets was
better: a diet of meat and
wine, or vegetables.
So he proposed that some
servants eat one diet and
the rest eat the other.
Then at the end of 10
days, they’d see who
looked healthier.
Controlled Experiments
In a controlled experiment there are two groups
who get separate treatments.
One group, the “control group” gets the
standard treatment. For example, all of the
king’s servants ate meat and wine before Daniel
suggested a different diet might be better.
Controlled Experiments
The other group, the “experimental group”, gets
the treatment we plan to test.
If the test group has better results than the
control group, we have good evidence that our
new treatment should be adopted.
Why are They Better?
Observational studies only reveal correlations,
they can’t reveal causation.
Controlled experiments are also only studies of
correlation: correlation between the control
group and outcomes, and correlation between
the experimental group and outcomes.
Why are They Better?
But controlled experiments are better than
observational studies. Why?
In observational studies, people are not
randomly assigned to conditions. For example,
an observational study might find a correlation
between using a cane and dying within a year.
Canes
This is because old people
are more likely to use a
cane and more likely to
die (than young people).
If you randomly assigned
young and old people to
cane or no-cane
conditions, the correlation
would go away. Canes
don’t cause death.
Confounding Variables
A confounding variable is a variable that affects
the variables you want to study.
For instance, if you want to study whether canes
cause death, age is a confounding variable,
because age influences your chances of death.
Confounding Variables
A controlled experiment lets you “control for”
confounding variables. You can make the control
group and the experimental group have equal
numbers of people from each age group.
Then you know that if more people in your
experimental group die, it wasn’t due to their
age (the other group had similar ages).
Controlling
In an observational study, there is no way to rule
out a common cause for two correlated
variables A and B.
In an experimental study, the common cause is
ruled out, because the experimenter is the one
who causes (“controls”) whether people have A
or not.
Controlling
In an observational study, there is no way to rule
out B causing A rather than A causing B. Does
wine reduce the risk of cancer, or does a
lowered risk of cancer increase wine
consumption?
If experimenters control who gets wine, then we
can rule out the hypothesis that in our study,
lowered cancer risk causes wine drinking.
Next Time
We’ll talk more next time about other things
that can bias an experiment and how to “control
for” them.
What about Observational Studies?
Why do scientists still conduct observational
studies, if controlled experiments are considered
better evidence?
1. Moral reasons
2. Practical reasons
Moral Reasons
Sometimes performing a controlled experiment
would be unethical.
For example, suppose we want to know whether
vaccines cause autism (NOTE: they do not).
We cannot simply stop vaccinating people.
Moral Reasons
If you stopped vaccinating
children, you’d effectively
be killing lots of children
(and adults).
Vaccines prevent lots and
lots of otherwise deadly
infectious diseases.
Moral Reasons
Thus you must conduct an observational study.
Find people who (for whatever reason) chose
not to vaccinate their children, and compare
their rates of autism to those of the vaccinated
children.
When you do this you find that vaccines do not
cause autism. (No correlation, hence no
causation.)
Practical Reasons
Some controlled experiments are also simply
impractical.
Does being smart make you rich? Well, we can’t
make a random group of people smart. That’s
impossible. Does being rich make you smart?
Well, we can’t give a random bunch of people a
lot of money– we’re just poor scientists!