Understanding Conditional Probability with an Example Video Lecture - (Maths)

You've tested positive for a rare and deadly
cancer that afflicts 1 out of 1000 people,
based on a test that is 99% accurate. What
are the chances that you actually have the
cancer? By the end of this video, you'll be
able to answer this question!
This video is part of the Probability and
Statistics video series. Many natural and
social phenomena are probabilistic in nature.
Engineers, scientists, and policymakers often
use probability to model and predict system
behavior.
Hi, my name is Sam Watson, and I'm a graduate
student in mathematics at MIT.
Before watching this video, you should be
familiar with basic probability vocabulary
and the definition of conditional probability.
After watching this video, you'll be able
to: Calculate the conditional probability
of a given event using tables and trees; and
Understand how conditional probability can
be used to interpret medical diagnoses.
Suppose that in front of you are two bowls,
labeled A and B. Each bowl contains five marbles.
Bowl A has 1 blue and 4 yellow marbles. Bowl
B has 3 blue and 2 yellow marbles.
Now choose a bowl at random and draw a marble
uniformly at random from it. Based on your
existing knowledge of probability, how likely
is it that you pick a blue marble? How about
a yellow marble?
Out of
the 10 marbles you could choose from, 4 are
blue. So the probability of choosing a blue
marble is 4 out of 10.
There are 6 yellow marbles out of 10 total,
so the probability of choosing yellow is 6
out of 10.
When the number of possible outcomes is finite,
and all events are equally likely, the probability
of one event happening is the number of favorable
outcomes divided by the total number of possible
outcomes.
What if you must draw from Bowl A? What's
the probability of drawing a blue marble,
given that you draw from Bowl A?
Let's go back to the table and consider only
Bowl A. Bowl A contains 5 marbles of which
1 is blue, so the probability of picking a
blue one is 1 in 5.
Notice the probability has changed. In the
first scenario, the sample space consists
of all 10 marbles, because we are free to
draw from both bowls.
In the second scenario, we are restricted
to Bowl A. Our new sample space consists of
only the five marbles in Bowl A. We ignore
these marbles in Bowl.
Restricting our attention to a specific set
of outcomes changes the sample space, and
can also change the probability of an event.
This new probability is what we call a conditional
In the previous example, we calculated the
conditional probability of drawing a blue
marble, given that we draw from Bowl A.
This is standard notation for conditional
probability. The vertical bar ( | ) is read
as "given." The probability we are looking
for precedes the bar, and the condition follows
the bar.
Now let's flip things around. Suppose someone
picks a marble at random from either bowl
A or bowl B and reveals to you that the marble
drawn was blue. What is the probability that
the blue marble came from Bowl A?
In other words, what's the conditional probability
that the marble was drawn from Bowl A, given
that it is blue? Pause the video and try to
work this out.
Going back to the table, because we are dealing
with the condition that the marble is blue,
the sample space is restricted to the four
blue marbles.
Of these four blue marbles, one is in Bowl
A, and each is equally likely to be drawn.
Thus, the conditional probability is 1 out
of 4.
Notice that the probability of picking a blue
marble given that the marble came from Bowl
A is NOT equal to the probability that the
marble came from Bowl A given that the marble
was blue. Each has a different condition,
so be careful not to mix them up!
We've seen how tables can help us organize
our data and visualize changes in the sample
space.
Let's look at another tool that is useful
for understanding conditional probabilities
- a tree diagram.
Suppose we have a jar containing 5 marbles;
2 are blue and 3 are yellow. If we draw any
one marble at random, the probability of drawing
a blue marble is 2/5.
Now, without replacing the first marble, draw
a second marble from the jar. Given that the
first marble is blue, is the probability of
drawing a second blue marble still 2/5?
NO, it isn't. Our sample space has changed.
If a blue marble is drawn first, you are left
with 4 marbles; 1 blue and 3 yellow.
In other words, if a blue marble is selected
first, the probability that you draw blue
second is 1/4. And the probability you draw
yellow second is 3/4.
Now pause the video and determine the probabilities
if the yellow marble is selected first instead.
If a yellow marble is selected first, you
are left with 2 yellow and 2 blue marbles.
There is now a 2/4 chance of drawing a blue
marble and a 2/4 chance of drawing a yellow
marble.
What we have drawn here is called a tree diagram.
The probability assigned to the second branch
denotes the conditional probability given
that the first happened.
Tree diagrams help us to visualize our sample
space and reason out probabilities.
We can answer questions like "What is the
probability of drawing 2 blue marbles in a
row?" In other words, what is the probability
of drawing a blue marble first AND a blue
marble second?
This event is represented by these two branches
in the tree diagram.
We have a 2/5 chance followed by a 1/4 chance.
We multiply these to get 2/20, or 1/10. The
probability of drawing two blue marbles in
a row is 1/10.
Now you do it. Use the tree diagram to calculate
the probabilities of the other possibilities:
blue, yellow; yellow, blue; and yellow, yellow.
The probabilities each work out to 3/10. The
four probabilities add up to a total of 1,
as they should.
What if we don't care about the first marble?
We just want to determine the probability
that the second marble is yellow.
Because it does not matter whether the first
marble is blue or yellow, we consider both
the blue, yellow, and the yellow, yellow paths.
Adding the probabilities gives us 3/10 + 3/10,
which works out to 3/5.
Here's another interesting question. What
is the probability that the first marble drawn
is blue, given that the second marble drawn
is yellow?
Intuitively, this seems tricky. Pause the
video and reason through the probability tree
with a friend.
Because we are conditioning on the event that
the second marble drawn is yellow, our sample
space is restricted to these two paths: P(blue,
yellow) and P(yellow, yellow).
Of these two paths, only the top one meets
our criteria - that the blue marble is drawn
first.
We represent the probability as a fraction
of favorable to possible outcomes. Hence,
the probability that the first marble drawn
is blue, given that the second marble drawn
is yellow is 3/10 divided by (3/10 +3/10),
which works out to 1/2.
I hope you appreciate that tree diagrams and
tables make these types of probability problems
doable without having to memorize any formulas!
Let's return to our opening question. Recall
that you've tested positive for a cancer that
afflicts 1 out of 1000 people, based on a
test that is 99% accurate.
More precisely, out of 100 test results, we
expect about 99 correct results and only 1
incorrect result.
Since the test is highly accurate, you might
conclude that the test is unlikely to be wrong,
and that you most likely have cancer.
But wait! Let's first use conditional probability
to make sense of our seemingly gloomy diagnosis.
Now pause the video and determine the probability
that you have the cancer, given that you test
positive.
Let's use a tree diagram to help with our
calculations.
The first branch of the tree represents the
likelihood of cancer in the general population.
The probability of having the rare cancer
is 1 in 1000, or 0.001. The probability of
having no cancer is 0.999.
Let's extend the tree diagram to illustrate
the possible results of the medical test that
is 99% accurate.
In the cancer population, 99% will test positive
(correctly), but 1% will test negative (incorrectly).
These incorrect results are called false negatives.
In the cancer-free population, 99% will test
negative (correctly), but 1% will test positive
(incorrectly). These incorrect results are
called false positives.
Given that you test positive, our sample space
is now restricted to only the population that
test positive. This is represented by these
two paths.
The top path shows the probability you have
the cancer AND test positive. The lower path
shows the probability that you don't have
cancer AND still test positive.
The probability that you
actually do have the cancer, given that you
test positive, is (0.001*0.99)/((0.001*0.99)+(0.999*0.01)),
which works out to about 0.09 - less than
10%!
The error rate of the test is only 1 percent,
but the chance of a misdiagnosis is more than
90%! Chances are pretty good that you do not
actually have cancer, despite the rather accurate
test. Why is this so?
The accuracy of the test actually reflects
the conditional probability that one tests
positive, given that one has cancer.
But in practice, what you want to know is
the conditional probability that you have
cancer, given that you test positive! These
probabilities are NOT the same!
Whenever we take medical tests, or perform
experiments, it is important to understand
what events our results are conditioned on,
and how that might affect the accuracy of
our conclusions.
In this video, you've seen that conditional
probability must be used to understand and
predict the outcomes of many events. You've
also learned to evaluate and manage conditional
probabilities using tables and trees.
We hope that you will now think more carefully
about the probabilities you encounter, and
consider how conditioning affects their interpretation.