Understanding Bayes Theorem
Random Signals and Noise (EE 326)
Introduction
The purpose of this assignment is to understand Bayes’ Theorem and to use it to find conditional probabilities of
events. Additionally, we will verify the results of applying Bayes’ Theorem by computer simulation, showing that
Bayes’ Theorem is in fact accurate, despite the counter-intuitive results that it sometimes gives. Bayes’ Theorem is
defined as:
p(A|B) = p(B|A)p(A)
p(B)(1)
Problem 1: The Monty Hall problem
Suppose you are a game show contestant, and you presented with three doors. A “prize” is behind each door, and you
will be given the prize that is behind the door that you choole. Behind one of these doors is a new car; goats are behind
the other two. The probability of the car being behind any given door is uniform and equal to 1
3After choosing a door,
the host of the game show reveals one of the other doors. Because the host knows where the car is, he will always
reveal a goat. You are asked if you want to change your selection to the other remaining door.
(a.) Without doing any computations, do you believe that choosing to switch/stay makes any difference? If so what do
you choose?
(b.) Using Bayes’ Theorem, compute the probability that there is a car behind the door that you chose. (Hint: Using
SC to represent the event that you have selected a car and GR to represent the event of a goat being revealed, you
are computing p(SC |GR).) Using the total probability theorem, compute the probability that the car is behind the
remaining door. Does this change your answer?
(c.) Suppose that for some reason, the host does not have knowledge of the car’s location and randomly chooses
to reveal one of the remaining doors. Supposing a goat is revealed, compute the probability that the door you selected
contains a car and the probability that the remaining door contains a car. Does it matter which door you pick? If so,
which one do you pick? (Hint: p(GR)has changed)
(d.) Verify your results in parts b and c by simulation. In both cases, use a sample size of 1,000, and randomly
select which door the car is behind and which door you initially choose.
In the case of part b, if the car is selected, have the host randomly choose reveal one of the remaining doors. If the car
is not selected, have the host choose the remaining door that contains the goat. Record the number of times the car is
behind the chosen door and the number of times it is behind the remaining door. Does this simulation support your
findings?
In the case of part c, have the host randomly choose reveal one of the remaining doors. Record the number of times
the car is behind the chosen door, the number of times it is behind the remaining door, and the number of times that
the car is revealed. Does this simulation support your findings?
Problem 2: Testing for a Disease
Suppose that there exists a disease that affects 1% of a population. A test exists to detect this disease. If an individual
has the disease, the test will return a positive result with probability .95. When applied to individuals who do not have
the disease, this test will return a positive result with probability .02.
(a.) Without doing any computations, make a guess at the probability that someone has the disease given that they
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