Probability Overview (1A)Mar 05, 2018 · Overview (1A) 5 Young Won Lim 3/5/18 Conditional...
Transcript of Probability Overview (1A)Mar 05, 2018 · Overview (1A) 5 Young Won Lim 3/5/18 Conditional...
![Page 1: Probability Overview (1A)Mar 05, 2018 · Overview (1A) 5 Young Won Lim 3/5/18 Conditional Probability Examples (2) 0, 0, 0, 0 0, 0, 0, 1 0, 0, 1, 0 0, 0, 1, 1 0, 1, 0, 0 0, 1, 0,](https://reader033.fdocuments.us/reader033/viewer/2022060915/60a90b86782bf354854ddda4/html5/thumbnails/1.jpg)
Young Won Lim3/5/18
Probability Overview (1A)
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Young Won Lim3/5/18
Copyright (c) 2017 - 2018 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using LibreOffice and Octave.
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Overview (1A) 3 Young Won Lim3/5/18
Conditional Probability
https://en.wikipedia.org/wiki/Conditional_probability
Given two events A and B with P(B) > 0, the conditional probability of A given B is defined as the quotient of the probability of the joint of events A and B, and the probability of B:
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Overview (1A) 4 Young Won Lim3/5/18
Conditional Probability Examples (1)
https://en.wikipedia.org/wiki/Conditional_probability
The unconditional probability P(A) = 0.52. the conditional probability P(A|B1) = 1, P(A|B2) = 0.75, and P(A|B3) = 0.
P (A∩B1) = 0.1P (A∩B2) = 0.12P (A∩B3) = 0
P (B1) = 0.1P (B2) = 0.16P (B3) = 0.1
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Overview (1A) 5 Young Won Lim3/5/18
Conditional Probability Examples (2)
0, 0, 0, 00, 0, 0, 10, 0, 1, 00, 0, 1, 10, 1, 0, 00, 1, 0, 10, 1, 1, 00, 1, 1, 11, 0, 0, 01, 0, 0, 11, 0, 1, 01, 0, 1, 11, 1, 0, 01, 1, 0, 11, 1, 1, 01, 1, 1, 1
0, 0, 0, 00, 0, 0, 10, 0, 1, 00, 0, 1, 10, 1, 0, 00, 1, 0, 10, 1, 1, 00, 1, 1, 1
0, 0, 0, 00, 0, 0, 10, 0, 1, 00, 0, 1, 10, 1, 0, 00, 1, 0, 10, 1, 1, 00, 1, 1, 11, 0, 0, 01, 0, 0, 11, 0, 1, 01, 0, 1, 11, 1, 0, 01, 1, 0, 11, 1, 1, 01, 1, 1, 1
P(E) =816
P(F ) =816
0, 0, 0, 00, 0, 0, 10, 0, 1, 00, 0, 1, 10, 1, 0, 00, 1, 0, 10, 1, 1, 00, 1, 1, 1
P(E∩F) =516
P(E∣F) =58
P(E∩F )
P (F)=
5 /168 /16
S E F E∩F
sample space E : at least two consecutive zero’s
F : starting with a zero
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Overview (1A) 6 Young Won Lim3/5/18
Intersection Probability
https://en.wikipedia.org/wiki/Conditional_probability
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Overview (1A) 7 Young Won Lim3/5/18
Independence
https://en.wikipedia.org/wiki/Independence_(probability_theory)
P (A∣B)= P(A)
P (B∣A)= P(B)
two events are (statistically) independent if the occurrence of one does not affect the probability of occurrence of the other.
Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.
H
H
H H
1/2
1/2
1 /4 = (1 /2)(1/2)
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Overview (1A) 8 Young Won Lim3/5/18
Independence
https://en.wikipedia.org/wiki/Independence_(probability_theory)
P (A∣B)= P(A)
P (B∣A)= P(B)P(A∩B) = P(A)P(B)
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Overview (1A) 9 Young Won Lim3/5/18
Coin Tossing Experiment (1)
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
3 H’s
2 H’s
2 H’s
1 H’s
2 H’s
1 H’s
1 H’s
0 H’s
0 T’s
1 T’s
1 T’s
2 T’s
1 T’s
2 T’s
2 T’s
3 T’s
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
Counting heads only Counting tails only
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Overview (1A) 10 Young Won Lim3/5/18
Coin Tossing Experiment (2)
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
3 H’s
2 H’s
2 H’s
1 H’s
2 H’s
1 H’s
1 H’s
0 H’s
0 T’s
1 T’s
1 T’s
2 T’s
1 T’s
2 T’s
2 T’s
3 T’s
H H H
H H T
H H
H T T
H H
H T
TH
T T T
T
T
T
T
Counting heads only determines tails
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Overview (1A) 11 Young Won Lim3/5/18
Coin Tossing Experiment (3)
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
3 H’s
2 H’s
1 H’s
0 H’s
0 T’s
1 T’s
2 T’s
3 T’s
H H H
H H T
H T T
T T T
× 3
× 3
× 1
× 1
Consider a combinationNot a permutation
C (3,3)
C (3,2)
C (3,1)
C (3,0)
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Overview (1A) 12 Young Won Lim3/5/18
Coin Tossing Experiment (4)
H H H
H H T
H T H
H T T
T H H
T H T
T T H
T T T
3 H’s
2 H’s
1 H’s
0 H’s
0 T’s
1 T’s
2 T’s
3 T’s
H H H
H H T
H T T
T T T
× 3
× 3
p p p
p p q
p q q
q q q
p3⋅C (3,3)
p2q⋅C (3,2)
p q2⋅C (3,1)
q3⋅C (3,1)
× 1
× 1
p3
p2q
p2q
p2q
p q2
p q2
p q2
q3
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Overview (1A) 13 Young Won Lim3/5/18
Random Variable is a function
● HHH● HHT● HTH● HTT● THH● THT● TTH● TTT
● 3
● 2
● 1
● 0
S: Sample Space R: Real Number
X(HHH) = 3
X(HHT) = 2
X(HTH) = 2
X(HTT) = 1
X(THH) = 2
X(THT) = 1
X(TTH) = 1
X(TTT) = 0
X(s)
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Overview (1A) 14 Young Won Lim3/5/18
Random Variable is related to events
X(HHH) = 3
X(HHT) = 2
X(HTH) = 2
X(HTT) = 1
X(THH) = 2
X(THT) = 1
X(TTH) = 1
X(TTT) = 0
A random variable does not return a probability.
X = 3
X = 2
X = 1
X = 0
Event { HHH }
Event { HHT, HTH, THH }
Event { HTT, THT, TTH }
Event { TTT }
looks like variables
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Overview (1A) 15 Young Won Lim3/5/18
Distribution
X(HHH) = 3
X(HHT) = 2
X(HTH) = 2
X(HTT) = 1
X(THH) = 2
X(THT) = 1
X(TTH) = 1
X(TTT) = 0
A random variable does not return a probability.
p(X = 3)
p(X = 2)
p(X = 1)
p(X = 0)
p(Event { HHH })
p(Event { HHT, HTH, THH })
p(Event { HTT, THT, TTH })
p(Event { TTT })
P3 = p3⋅C (3,3)
P2 = p2q⋅C (3,2)
P1 = pq2⋅C (3,1)
P0 = q3⋅C (3,1)
{ (0 , P0) , (1 ,P1) , (2 , P2) , (3 , P3) }Distribution
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Overview (1A) 16 Young Won Lim3/5/18
A random variable and its distribution
● HHH● HHT● HTH● HTT● THH● THT● TTH● TTT
● 3
● 2
● 1
● 0
S: Sample Space R: Real Number
X(s)
● 1/8
● 3/8
R: Real Number
p(X)
Random variable Distribution
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Overview (1A) 17 Young Won Lim3/5/18
Different Random Variable Assignments
0 1 2 3
18
38
38
18
38
38
18
18
1 2 3 4X
p (X)
X
p (X)
3 H’s
2 H’s
1 H’s
0 H’s
H H H
H H T
H T T
T T T
3 H’s
2 H’s
1 H’s
0 H’s
H H H
H H T
H T T
T T T
X = 3
X = 2
X = 1
X = 0
X = 4
X = 1
X = 2
X = 3
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Overview (1A) 18 Young Won Lim3/5/18
Different Expectation Values
0 1 2 3
18
38
38
18
38
38
18
18
1 2 3 4X
p (X)
X
p (X)
E(X) = 0⋅18
+ 1⋅38
+ 2⋅38
+ 3⋅18
= 1.5
E(X) = 1⋅38
+ 2⋅38
+ 3⋅18
+ 4⋅18
= 2
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Overview (1A) 19 Young Won Lim3/5/18
Normal Distribution
https://en.wikipedia.org/wiki/Normal_distribution
Probability mass function Cumulative distribution function
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Overview (1A) 20 Young Won Lim3/5/18
Binomial Distribution
https://en.wikipedia.org/wiki/Binomial_distribution
Probability mass function Cumulative distribution function
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Overview (1A) 21 Young Won Lim3/5/18
Binomial Distribution
https://en.wikipedia.org/wiki/Bernoulli_trial
the binomial distribution with parameters n and p
the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome:
a random variable containing single bit of information: success / yes / true / one (with probability p) failure / no / false / zero (with probability q = 1 − p).
A single success / failure experimenta Bernoulli trial or Bernoulli experiment
a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution.
a sequence of outcomesa Bernoulli process
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Overview (1A) 22 Young Won Lim3/5/18
Binomial Distribution
https://en.wikipedia.org/wiki/Bernoulli_trial
The binomial distribution is
the basis for the popular binomial test of statistical significance.
frequently used to model the number of successesin a sample of size n drawn with replacement from a population of size N.
the sampling carried out without replacementthe draws are not independenta hypergeometric distributionnot a binomial distribution
for N much larger than n, the binomial distribution remains
a good approximationwidely used.
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Overview (1A) 23 Young Won Lim3/5/18
Bernoulli Trial
https://en.wikipedia.org/wiki/Bernoulli_trial
a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.
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Overview (1A) 24 Young Won Lim3/5/18
Dependent Events
https://en.wikipedia.org/wiki/Bernoulli_trial
Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.
Blue arrow: Throwing a 6-sided dice 6 times gives 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0.
Grey arrow: To get 50-50 chance of throwing a Yahtzee (5 cubic dice all showing the same number) requires 0.69 × 1296 ~ 898 throws.
Green arrow: Drawing a card from a deck of playing cards without jokers 100 (1.92 × 52) times with replacement gives 85.7% chance of drawing the ace of spades at least once.
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Overview (1A) 25 Young Won Lim3/5/18
Tossing Coins Probability
https://en.wikipedia.org/wiki/Bernoulli_trial
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Overview (1A) 26 Young Won Lim3/5/18
Binomial Distribution Examples
Binomial distribution for p = 0.5 with n and k as in Pascal's triangle
The probability that a ball in a Galton box with 8 layers (n = 8) ends up in the central bin (k = 4) is 70 / 256
https://en.wikipedia.org/wiki/Binomial_distribution
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Overview (1A) 27 Young Won Lim3/5/18
Bean Machine
https://en.wikipedia.org/wiki/Bean_machine
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Overview (1A) 28 Young Won Lim3/5/18
Binomial Distribution – Mean
https://en.wikipedia.org/wiki/Binomial_distribution
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Overview (1A) 29 Young Won Lim3/5/18
Binomial Distribution – Variance
https://en.wikipedia.https://en.wikipedia.org/wiki/Binomial_distribution/wiki/Algorithm
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Overview (1A) 30 Young Won Lim3/5/18
Central Limit Theorem
https://en.wikipedia.org/wiki/Central_limit_theorem#/media/File:Central_limit_thm.png
A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations;
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Overview (1A) 31 Young Won Lim3/5/18
Central Limit Theorem
https://en.wikipedia.org/wiki/Central_limit_theorem
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Young Won Lim3/5/18
References
[1] http://en.wikipedia.org/[2] https://en.wikiversity.org/wiki/Discrete_Mathematics_in_plain_view