Slides 5: Reasoning with Random Numbers

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1 Slides 5: Reasoning with Random Numbers Realistic Simulating What is convincing? Fast and efficient? Solving Problems Simulation Approach Using Thought Experiment or Theory Formal Language 1

2 Brain Teasers Who is the murderer? Perpetrator has blood type A. Perpetrator may be known person x or unknown other y. Equal probability in absence of other information. 10% of population have blood type A. Suspect x has blood type A. What is the probability x is the perpetrator given the available information? 2

3 Spreadsheet Model 3

4 Another Example One in a 1,000 people suffer from rare disease D. A test for disease D is reliable 90% of the time, i.e., gives the correct result with probability 0.9. I take the test and it comes back positive, what is the probability I have the disease? 4

5 Real Experiences Real experiences: Compare with the case of Sally Clarke 5

6 Brain Teasers: Mini-League Teams A, B and C. Know C was not outright winner. What is the probability that A is outright winner? 6

7 Mini-League In these simulations compute: Proportion of times A is outright winner (OW) Proportion of times A score is greater than B score If I know A = 1 what is proportion of times B = 1? If I know C not OW, then proportion A is OW, or equivalently, proportion of times A score larger than B score. 7

8 Mini-League 8

9 Mini-League In these simulations compute: Proportion of times A is outright winner (OW) A OW if and only if A scores 2 (236 out of 1000) Proportion of times A score is greater than B score A > B if and only if (1,0,2),(2,0,1),(2,1,0) ( out of 1000) If I know A = 1 what is proportion of times B = 1 (252 out of 499) If I know C not OW, then proportion A is OW, or equivalently, proportion of times A score larger than B score 236/( ). 9

10 Mini-League 10

11 Long Run: Mini-League In these simulations compute: Probability A is outright winner (OW)? Probability A score is greater than B score? If I know A = 1 what is probability B = 1? If I know C not OW, what is probability A is OW, or equivalently, that A score is larger than B score? What about expected totals when all teams have equal skill, or if A is twice as good as the others? 11

12 Mini-League 12

13 Brain Teasers Monty Hall Game One car behind one of three doors, the other two have a goat behind them. Player selects one, say Door 1. Before opening this door, the host (who knows what is behind each door), opens one of the other two doors, say door 2, and shows a goat. Host now offers chance to change selection. Issue: Is there any point in changing? 13

14 Monty Hall Game Show Policy 1: Always switch Policy 2: Never switch What is random? which door has car which door host opens which door to choose is a policy, so not random Simulation (conditional) 14

15 Spreadsheet Model 15

16 Back to System Redundancy 16

17 Changing Sigma 17

18 System Redundancy 18

19 Force of Mortality 19

20 Force of Mortality 20

21 Force of Mortality 21

22 Random Walk System: at time t, X t ; at time t+1, X t+1. change in (t,t+1), D t, random, independent and identically generated. X t+1 = X t +D t Model Current location of: Drunkard Photon emerging from Sun Accumulated: Winnings in series of games. Profit from investment. Long Run average 1/n times the sum over 1 to n. 22

23 Can you beat the bank? Game, with fixed probability p of winning. Initial pot $100, are you going to win in the long run? 23

24 Gambler s Ruin Game, with fixed probability p of winning and initial pot $100. What happens if you continue until you are out of money or have won a stated amount? From theory - for even chance of win or lose - probability that the bank ruined under the assumption that you start with a and bank starts with b is a a+b 24

25 Kelly Betting Game, with fixed probability p of winning. Initial pot $100, payout is f times the bet. Strategy 1, fixed bet. Strategy 2, bet a fixed fraction of the pot. 25

26 Kelly Betting - what fraction? 26

27 Betting Strategy Fixed bet implies ruin eventually. Kelly bet implies ruin never. From theory the optimal fraction to bet is α = pf 1 f 1 of win and f is payout odds. where p is probability A Fractional Kelly is cα for c < 1. 27

28 Day Trading: Multiplicative Random Walk Invest X 0 = $10 Fixed daily RoR of 0.75% After n days X n = X n 1 (1+RoR/100) So X n = 10( ) n. For variable RoR n after n days X n = X n 1 (1+RoR n /100) This is additive on the log scale as log(x n ) = log(x n 1 )+log(1+ror n /100) 28

29 Day Trading: Multiplicative Random Walk 29

30 Mathematical Approach Thought experiments: If one were to simulate/replicate/summarise what would be the long run outcome? Math models: Similar solutions for many problems. For example the birthday problem is equivalent to duplications in passwords. Math notation - shorthand! Probability that, given information, A happens is P(A information). The Key is conditional probability! 30

31 Two Queens: P(Q 1 Q 2 ) Symmetry: Thought experiment in N replications: In N replications find Q 2 in (4/52)N. Of these find Q 1 in (4/52)(3/51)N. Proportion of Q 1 given Q 2 is (4/52)(3/51)N (4/52)N = 3/51 So long run P(Q 1 Q 2 ) = 3/51. 31

32 Who is the murderer? Thought experiment in N replications 32

33 Conditional, Joint, Marginal 33

34 Monty Hall Game Show Policy never switch: Win if and only if first door has a car. Symmetry implies P(win under never) = 1/3 Policy always switch: Win if and only if first door does not have a car. Symmetry implies P(win under always) = 2/3 P(Win host information) = 2/3 = 2 P(Win no host information) 34

35 Gambler s Ruin Thought experiment for symmetric case. You start with $a; opponent starts with $b; total $a+b. P(you win) = 1/2 = 1 P(you lose) P k = P(eventual ruin for bank you start with X 0 = k) = P(Bank ruin and you win or lose first game X 0 = k) = 1 2 P(Bank ruin win first X 0 = k)+ 1 2 P(Bank ruin and lose first X 0 = k) = 1 2 P(Bank ruin X 0 = k +1)+ 1 2 P(Bank ruin X 0 = k 1) So P k = 1 2 P k P k 1 for k = 1,2,...,a+b 1 Note P 0 = 0 and P a+b = 1 35

36 Gambler s Ruin Thought experiment for symmetric case. P k = 1 2 P k P k 1 for k = 1,2,...,a+b 1 and P 0 = 0 and P a+b = 1. From recursion: P k+1 = 2P k P k 1 P 2 = 2P 1 0 etc. implies P 3 in terms of P 1 But P a+b = 1 and solve for P 1 Hence P a = P(bank ruined you start with a bank with b) = a a+b 36

37 Real Problems What is the problem? Simulate system (for how long?) Summarise (how?) Consider subset of results? 37

38 Bayesian Spam Filtering 38

39 Benford s Law 39

40 Mathematics Probability rules for events Random Variables discrete or continuous univariate or multivariate Probability Distributions discrete or continuous; cumulative or not summaries of many samples is equivalent to the probability distribution. 40

Thinking with Probability

Thinking with Probability Thinking fast and slow Thinking by decomposing events formalising probability rules conditional probability Brain Teasers Tijms Problems, Chap 1 2 dice rolled; one is 6. What