Randomness, Probability Basics, PMFs

Randomness, Probability Basics, PMFs
Submit a PDF of your answers to Canvas.
1. Laws of Probability. Prove or disprove if the following statements are true in general:
a) P(Ac
, Bc
) = P(Ac
) − P(B)
b) P(Ac ∩ Bc
) ≥ P(Ac
) − P(B)
c) P(A) − P(B) ≤ P(Bc
) − P(Ac
)
2. The Monty Hall Paradox (Adapted from Vox Savant, 1990). You’re on a gameshow,
and given the choice of three doors, denoted A, B, and C. Behind one is gold, and
behind the other two is nothing. Imagine you a pick door. Monty Hall, the host,
knows what is behind each door, and opens one of the remaining two doors and shows
you there is nothing behind it. He then says to you - ‘Do you want pick another door?’
a) Before you make your decision to move, what is the sample space for this random
experiment? Hint: write out all possible outcomes of the triple (door with gold,
door selected by player, door that Monty opens).
b) Assume that the gold was placed at random, and that your initial choice of door
is independent of the gold placement, and that Monty’s choice is random among
possible empty doors. Specify the probability measure for the outcomes, and use
it to compute the probability of winning if the contestant stays or moves.
3. Tertiary (3-class) Classification problem and crossover probabilities. You build a classifier that aims to classify pictures of cats, dogs and fish. The classifier succeeds at
classifying cats as cats with probability 1 − , but incorrectly labels cat pictures as
dogs with probability , and so on, as specified in the transition matrix below:
Pictures of cats occur with probability 1/2, pictures of dogs with probability 1/4, and
pictures of fish with probability 1/4.
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a) Find, as a function of , the probabilities of the labels.
b) Suppose the classifier predicts dog. What is the probability that the input was
cat? Fish? Dog? Express your answer a function of .
4. Multinomial Probabilities by Enumeration. The multinomial distribution is an extension of the binomial distribution. For n independent trials of an experiment that
leads to one of k possible categories (i.e, the roll of a k = 6 sided dice, n times), the
multinomial distribution models the probability of the counts of each category.
Imagine that you toss a fair, 6 sided dice 7 times.
a) Categorical outcomes. What is Ω in this experiment? Write a script to enumerate
the possible outcomes (note that you can do this with nested for loops, or a
recursive function). Your script should produce a list of tuples, where each tuple
has 7 elements indicating the sequence of results from tossing the dice seven times.
b) How many total outcomes are there? What is the probability of any given outcome?
c) Multinomial outcomes. Suppose you are only interested in the number of times
each of the categories (sides of the dice, 1-6) appeared in the 7 throws. This count
or frequency of each categories is a multinomial random variable. What is Ω for
this experiment?
d) Write a script to convert the outcomes from (a) into this format, keeping track
of how many outcomes in (a) map to a single multinomial outcome. It may be
useful to use a dictionary data structure.
e) How many possible multinomial outcomes are there?
f) What are the most probable multinomial outcomes, and what is their probability?
g) (Optional) Derive a general expression for the total number of multinomial outcomes given k categories and n trials.
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