Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to...
Transcript of Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to...
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The Probabilistic Method
Joshua BrodyCS49/Math59
Fall 2015
Week 1: Introduction to Probability Theory
![Page 2: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/2.jpg)
Clicker Question
(A) A probability distribution is a random variable.
(B) A probability distribution is a function.
(C) A probability distribution is an event.
(D) None of the above.
What is a probability distribution?
![Page 3: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/3.jpg)
Clicker Question
(A) A probability distribution is a random variable.
(B) A probability distribution is a function.
(C) A probability distribution is an event.
(D) None of the above.
What is a probability distribution?
![Page 4: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/4.jpg)
Probability Distributions
Let Ω be a finite, non-empty set. A probability distribution on Ω is a function P: Ω →{0,1} such that
∑ P(w) = 1w ∈ Ω
• Ω : sample space of P
• support of P: {w : P(w) > 0}
![Page 5: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/5.jpg)
Clicker Question
(A) Pr[k] = 1/n.
(B) Pr[k] = k/n.
(C) Pr[k] = /2n
(D) Pr[k] = k2/2n.
binomial distribution: flip a coin n times and count # heads.
What is Pr[k]?
nk
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![Page 6: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/6.jpg)
Events
![Page 7: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/7.jpg)
Events
![Page 8: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/8.jpg)
Example Problem
Arithmetic Sequence:
list of numbers (a1, a2, ..., am) where ai = ai-1 + k for some k
Problem: show how to color numbers {1, 2, 3, 4, ..., 2015} using 4 colors so that no arithmetic sequence is monochromatic
Solution: color each number 1, 2, ..., 2015 randomly.
examples: (1, 5, 9, 13), (201, 402, 603, 804, 1005)
![Page 9: Week 1: Introduction to Probability Theorybrody/cs49/f15/slides/CS...Week 1: Introduction to Probability Theory Clicker Question (A) A probability distribution is a random variable.](https://reader033.fdocuments.us/reader033/viewer/2022060304/5f092c957e708231d4259448/html5/thumbnails/9.jpg)
The Probabilistic Method