Math-UA.233: Theory of Probability
Lecture 14: review before the midterm
Tim Austin
tim@cims.nyu.edu
cims.nyu.edu/∼tim
The midterm cheatsheet on the class website breaks down the
definitions, theorems, facts and ideas that you need to know.
Let’s start here with a more compact sketch of how the course
material breaks down so far:
COUNTING :
PROBABILITY :
COND. PROB. :
RVs :
The basic principle of counting
Permutations, combinations, multinomial coeffs
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Experiments, sample spaces, outcomes, events
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 Manipulating events, De Morgan, etc.
Axioms of probability
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 Re-arranging and computing probabilites
Examples with equally likely outcomes
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Definition and basic calculations
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 The Multiplication Rule
The Law of Total Probability
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 Bayes’ formula
Independence: definition and use
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Definition
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 Indicator variables
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 Discrete RVs, their PMFs, calcs using the PMF
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Expectation and variance
E[g(X )]
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Linearity of Expectation, expected #s of things
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 Bernoulli and binomial RVs
Poisson RVs and Poisson approximation
COUNTING
Example (Ross Prob 1.16)
A student has to sell 2 books from a collection of 6 math, 7
science and 4 economics books. How many choices are
possible if
(a) both books are to be on the same subject?
(b) the books are to be on different subjects?
PROBABILITY, COUNTING
Example (Part of Ross Self-test 2.18)
Four red, eight blue and five green balls are randomly arranged
in a line. Find the probability that
(a) the first five balls in the line are blue.
(b) none of the first five balls is blue.
CONDITIONAL PROBABILITY
Example (Ross Prob 3.24)
Each of two balls is painted either black or gold and then placed
in an urn. Each ball is painted black with probability 12 and
these events are independent.
(a) Suppose that you obtain information that the gold paint has
been used (and so at least one ball is gold). Find the
conditional probability that both balls are painted gold.
(b) Now suppose in addition that you withdraw one of the
balls. What is the probability that it is gold?
(c) Finally, suppose the ball that you withdrew in part (b) is
gold. What is the probability that both balls are gold in this
case?
CONDITIONAL PROBABILITY (INDEPENDENCE)
Example (Adapted from Ross E.g. 4.8d)
Independent trials are performed. Each results in success with
probability p. Let E be the event that r successes occur before
m failures. Let F be the event that the r th success occurs on or
before the (r + m − 1)th trial.
(a) Show carefully that E = F .
(b) Find P(E ).
PROBABILITY, RANDOM VARIABLES
Example (Ross Prob 2.33)
A forest contains 20 elk, of which 5 are captured, tagged and
then released. A certain time later, 4 of the 20 elk are captured.
(a) What is the probability that 2 of these 4 have been tagged?
What are you assuming about the probabilities about
capturing different elk?
(b) Find the expected number of tagged elk among the second
sample.
Example (Adapted from Ross self-test 4.14)
On average, Gondwanaland suffers 5.2 hurricanes per year.
Small weather systems are much more common, but each of
them has only a small independent probability of turning into a
hurricane.
(a) What is the probability that there will be three or fewer
hurricanes in the next year? State any modeling
assumptions.
(b) Give a formula for the probability that there will be six or
fewer hurricanes in the next two years. Is it greater or less
than the square of the answer to part (a)? [Hint: what
event that concerns the next two years would have
probability equal to the square of the answer to part (a)?]
CONDITIONAL PROBABILITY, RANDOM VARIABLES
Example (From Ross self-test 4.13)
Shane asks five separate fortune tellers whether Constanza
stole his phone. Each of the fortune tellers predicts correctly
with probability 0.7. Given that three of them say Constanza is
guilty, what is the probability that Constanza is actually guilty?