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# Probability+posted - Probability some concepts Experiment:...

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E7 L20 1 Probability some concepts Probability some concepts Experiment : A situation whose outcome depends on chance Sample Space S The set of all possible outcomes of an experiment Example Experiment: throwing of a dice once S =

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E7 L20 2 Probability - events Probability - events Experiment: throwing of a dice once S = Outcomes : elements of the sample space S Events : Are subsets of the sample space S Empty subsets are null or impossible events An event occurs if any of the outcomes in that event occurs.
E7 L20 3 Probability - events Probability - events Experiment: throwing of a dice once Some events: The event of observing an even number of dots: The event O of observing an odd number of dots: S = E = O =

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E7 L20 4 Probability - Sample space, outcome, event Probability - Sample space, outcome, event Experiment: throwing a pair of dice (one red and one blue ) the sample space has 36 outcomes: S = The event L of obtaining the number 7 is { } ( , ),( , ),( , ),( , ),( 1 2 3 4 5 6 5 4 3 2 , ),( ) 1 6 , L = occurs if any of the outcomes in occurs.
Probability Probability A number between 0 and 1, inclusive, that indicates how likely an event is to occur . An event with probability of 0 is a null event. An event with probability of 1 is a certain event. Probability of event A is denoted as P(A) . The closer P(A)   to 1 , the more likely is A   to happen. E7 L20 5

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Probability Probability A number between 0 and 1, inclusive, that indicates how likely an event is to occur . An event with probability of 0 is a null event. the earth will stop spinning tomorrow getting a A in E7 without doing the labs An event with probability of 1 is a certain event. taxes will always exist in some form a person eventually dies E7 L20 6
Assigning Probability - frequentist approach Assigning Probability - frequentist approach An experiment is repeated n times under essentially identical conditions if the event A occurs m times, then as n grows large E7 L20 7 ( ) m P A n

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E7 L20 8 Dice example Dice example Experiment: throwing of a fair dice once S = { } 1,2,3,4,5,6 S = P(S) = 1 P(1) = 1/6,   P(3) =1/6, P(6) = 1/6  P(even number) = 3/6 = 1/2 P(odd number) = 3/6 = 1/2
Random numbers – matlab simulation Random numbers – matlab simulation Matlab rand function generates a uniformly distributed pseudo-random number between [0,1] >> X = rand E7 L20 9 ( 1) 1 P X ≤ = 3 3 ( ) 4 4 P X ≤ = 1 3 3 1 1 ( ) ( ) ( ) 4 4 4 4 2 P X P X P X ≤ ≤ = ≤ - ≤ = 0 0.5 1 0 0.2 0.4 0.6 0.8 1 x P(X < x) ( ) P X x

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Random numbers – matlab simulation Random numbers – matlab simulation Matlab rand function >> test_rand(100) ans = 1.0000 0.8100 0.5900 >> test_rand(1e6) ans = 1.0000 0.7504 0.4999 E7 L20 10 function P = test_rand(n) X = rand(n,1); P=zeros(1,3); P(1) = sum(X <= 1)/n; P(2) = sum(X <= 3/4)/n; P(3) = sum((1/4 <= X) & (X <= 3/4))/n; n is not large enough ( ) m P A n
Dice– matlab simulation Dice– matlab simulation Throw a fair dice n times >> [D,P] = dice1(1e6); >> P P = 0.1667 0.1669 0.1667 0.1663 0.1667 0.1666 E7 L20 11 function [D,P] = dice1(n) X = 6*rand(n,1); D = floor(X)+1; % Estimate probability of each outcome P = zeros(1,6); for i = 1:6 P(i) = sum(D==i)/n; end D(i)

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## Probability+posted - Probability some concepts Experiment:...

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