Bayes’ theorem can tell us the probability that some hypothesis is true given an event oc- currence. The theorem instruc

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Bayes’ theorem can tell us the probability that some hypothesis
is true given an event oc-
currence. The theorem instructs us to take the prior probability
that a hypothesis is true and
multiply it by the probability of an event given that the
hypothesis is true. Then divide this by
the total probability of the event occurring. It is usually written
in the following way:
P(H|E) = P(E|H)(P(H)
P(E)
Where Hstands for hypothesis and Estands for event. The vertical
pipe |between the Hand
Ecan be read as given. i.e. The left side of the equality can be
read as ”the probability of
the hypothesis given the event.” It is usually more useful to write
the term in the denominator,
P(E), as the following sum:
P(E) = P(E|H)P(H) + P(E|¬H)P(¬H)
Where the ¬Hindicates NOT H. This theorem will allow us to
calculate probabilities for events
that may occur in the laboratory. Let us consider a concrete
example. Suppose that there is
a bag containing 3 red chips and 7 green chips. A chip is then
drawn from the bag at random
by a person who is red-green colorblind. The person exclaims ”it’s
red!” after drawing the chip.
However, because of their colorblindness, they can only correctly
identify red from green 80% of
the time. We could then ask, what is the probability that this
person has actually drawn a red
chip? Applying Bayes’ theorem we find:
P(H|E) = P(E|H)P(H)
P(E|H)P(H) + P(E|¬H)P(¬H)
= (0.80)(3/10)
(0.80)(3/10) + (0.20)(7/10)
≈0.63
We find that the probability of it being a red chip is actually
63%. This is not likely what you
would have guessed. The results of Bayes’ theorem are often
surprising.
Suppose that you go to the doctor for a regular checkup and even
though you have no
symptoms, the doctor requests that you be tested for a rare
disease, a disease that only 1 in
10,000 people in your age group contract. The doctor informs you
that the test is 99% accurate,
meaning that, if you have the disease, it comes back positive 99%
of the time and incorrectly
produces a negative result, 1% of the time. After a week, the
results come back. The doctor
informs you that you have tested positive. (a) Use Bayes’ theorem
to calculate the probability
that you have the disease. (b) Suppose that after your positive
test, the doctor asks you to
retake the test. Again it comes back positive. What is the
probability that have the disease
now? (Hint: use the answer of part (a) as the new hypothesis) (c)
Suppose that instead of being
symptom free, you have symptoms that are consistent with the
disease. You are told by the
doctor that people in your age group with symptoms will have a 1 in
12 chance of contracting
the rare disease. If the test is positive, what is the probability
that you have the disease?