Probability Refresher and Cycle Analysis. Spring 2018 CS 438 Staff, University of Illinois 1

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1 Probability Refresher ad Cycle Aalysis Sprig 2018 CS 438 Staff, Uiversity of Illiois 1

2 A Quick Probability Refresher A radom variable, X, ca take o a umber of differet possible values Example: the umber of pigeos o the widowsill outside is a radom variable with possible values 1,2,3, Each time we observe (or sample) the radom variable, it may take o a differet value Sprig 2018 CS 438 Staff, Uiversity of Illiois 2

3 A Quick Probability Refresher A radom variable takes o each of these values with a specified probability Example: X = {0, 1, 2, 3, 4} P[X=0] =.1, P[X=1] =.2, P[X=2] =.4, P[X=3] =.1, P[X=4] =.2 The sum of the probabilities of all values equals 1 S all values P[X=value] = 1 Sprig 2018 CS 438 Staff, Uiversity of Illiois 3

4 A Quick Probability Refresher Example Suppose we throw two dice ad the radom variable, X, is the sum of the two dice Possible values of X are {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} P[X=2] = P[X=12] = 1/36 P[X=3] = P[X=11] = 2/36 P[X=4] = P[X=10] = 3/36 P[X=5] = P[X=9] = 4/36 P[X=6] = P[X=8] = 5/36 P[X=7] = 6/36 Note: 12 S i=2 P[X=i] = 1 Sprig 2018 CS 438 Staff, Uiversity of Illiois 4

5 A Quick Probability Refresher Expected Value Ca be thought of a log term average of observig the radom variable a large umber of times E[X] = x = S All possible values of x Value * P[X = value] Example: dice - E[X] = 2*1/36 + 3*2/36 + 4*3/36 + 5*4/36 + 6*5/36 + 7*6/36 + 8*5/36 + 9*4/ *3/ *2/ *1/36 Sprig 2018 CS 438 Staff, Uiversity of Illiois 7

6 Probability Example Basic probability otios Two useful rules Probabilities of all possible evets sum to 1 Probability of idepedet evets Product of probabilities of evets e.g., probability of two cois comig up heads = 1/2 x 1/2 = 1/4 Calculatig averages/expected values Fuctio f Multiply f by probability for each possible evet Sum over all evets Sprig 2018 CS 438 Staff, Uiversity of Illiois 10

7 Probability Example - Problem Give a bag with N balls 1 blue ball N - 1 white balls Algorithm pick a ball if blue, you wi else retur to bag repeat N times Questio What is your chace of wiig for large N? Sprig 2018 CS 438 Staff, Uiversity of Illiois 11

8 Probability Example - Solutio Ca write as a sum Chace of fidig blue o first try = 1/N O secod try = [(N-1)/N] * (1/N) Etc. Istead, write 1 - (chace of losig) Parethesized term Product of N factors Each factor = (N-1)/N 1 - [(N - 1)/N] N Sprig 2018 CS 438 Staff, Uiversity of Illiois 12

9 Probability Example - Solutio For N = 2, 1/2 first is white 1/2 secod is white 1/4 both are white 3/4 chace to wi = 1 - (1/2) 2 For N=3, 2/3 first is white 2/3 secod is white 2/3 third is white 8/27 all three are white 19/27 chace to wi = 1 - (2/3) 3 (< 3/4) Sprig 2018 CS 438 Staff, Uiversity of Illiois 13

10 Probability Example - Solutio N=4 probability of wi = 68% N=5 probability of wi = 67% N=8 probability of wi = 66% large N? 0? lim N æ N -1ö ç è N ø N Sprig 2018 CS 438 Staff, Uiversity of Illiois 14

11 Fu Example Flip a coi repeatedly. Two heads i a row scores 1 poit. Scorig pairs may ot overlap (e.g., three heads i a row does ot score 2 poits). O average, how may poits do you score per flip? Sprig 2018 CS 438 Staff, Uiversity of Illiois 15

12 A Differet Example What fractio of time (o average) is spet i state E? S A C B D E Sprig 2018 CS 438 Staff, Uiversity of Illiois 16

13 Cycle Aalysis Start with a discrete Markov process Trasitios happe periodically (every Dt) Probabilities idepedet of past/future behavior Form all possible cyclic sequeces (cycles) Pick a start state List all cycles from that state Calculate probability per cycle Calculate average cycle legth Ca calculate expected values of cycle-depedet properties with average legth ad cycle probabilities Sprig 2018 CS 438 Staff, Uiversity of Illiois 17

14 Example cycle probability S A C 0.25 B 0.75 D E Sprig 2018 CS 438 Staff, Uiversity of Illiois 18

15 Example cycle ABS CBS CDES probability = = 0.5 = = = = S A S A B B S A C 0.25 B 0.75 D E average cycle legth = = ABS CBS CDES Sprig 2018 CS 438 Staff, Uiversity of Illiois 19

16 Example Amout of time spet i E whe i cycle CDES average fractio of time spet i E S = periods/cycle dividig by average legth = / = A C 0.25 B 0.75 D Probability of cycle CDES E Sprig 2018 CS 438 Staff, Uiversity of Illiois 20

17 Fu Example Flip a coi repeatedly. Two heads i a row scores 1 poit. Scorig pairs may ot overlap (e.g., three heads i a row does ot score 2 poits). O average, how may poits do you score per flip? Sprig 2018 CS 438 Staff, Uiversity of Illiois 21

18 Fu Example cycle probability T 1/2 HT 1/4 HH 1/4 average cycle legth average score per cycle average score per flip T H (score) H oe 1 T Sprig 2018 CS 438 Staff, Uiversity of Illiois 22

19 Fu Example cycle probability T 1/2 HT 1/4 HH 1/4 average cycle legth average score per cycle average score per flip T H (score) H oe 1 T = 1/2 + 1/2 + 1/2 = 3/2 flips = 1/4 poits = (1/4) / (3/2) = 1/6 pts/flip Sprig 2018 CS 438 Staff, Uiversity of Illiois 23