Lec 18 Nov 12 Probability – definitions and simulation

1. Lec 18 Nov 12 Probability – definitions and simulation

2. (Discrete) Sample space • Experiment: a physical act such as tossing a coin or rolling a die. • Sample space – set of outcomes. • Coin toss Sample Space S = { head, tail} • Rolling a die Sample space S = {1, 2, 3, 4, 5, 6} • Tossing a coin twice. Sample space S = {(h,h), (h,t), (t,h), (t,t)}

3. Events and probability • Event E is any subset of sample space S. • You flip 2 coins • Sample space S = {(h,h), (h,t), (t,h), (t,t)} • Event: both tosses produce same result E = {(h,h), (t,t)} • Prob(E) = |E|/ |S| • In the above example, p(E) = 2/4 = 0.5 Question: what is the probability of getting at least one six in three roles of a die?

4. Bernoulli trial • Bernoulli trials are experiments with two outcomes. (success with prob = p and failure with prob = 1 – p.) • Example: rolling an unloaded die. Success is defined as getting a role of 1. p(success) = 1/6

5. Random Variable • Random variable (RV) is a function that maps the sample space to a number. • E.g. the total number of heads X you get if you flip 100 coins • Another example: • Keep tossing a coin until you get a head. The RV n is the number of tosses. • Event = { H, TH, TTH, TTTH, … } RV n(H) = 1, n(TH) = 2, n(TTH) = 3, … etc.

6. Common Distributions • Uniform X: U[1, N] • X takes values 1, 2, …, N • E.g. picking balls of different colors from a box • Binomial distribution • X takes values 0, 1, …, n • N coin tosses. What is the prob. That there are exactly k tails?

7. Conditional Probability • P(A|B) is the probability of event A given that B has occurred. • Suppose 6 coins are tossed. Given that there is at least one head, what is the probability that the number of heads is 3? • Definition: p(A|B) =

8. Baye’s Rule If X and Y are events, then p(X|Y) = p(Y|X) p(X)/p(Y) Useful in situation where p(X), p(Y) and p(Y|X) are easier to compute than p(X|Y).

9. Independent events • Definition: X and Y are independent if

10. Monty Hall Problem • You're given the choice of three doors: Behind one door is a car; behind the others, goats. You want to pick the car. • You pick a door, say No. 1 • The host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. • Do you want to pick door No. 2 instead?

12. Monty Hall Problem: Bayes Rule • : the car is behind door i, i = 1, 2, 3 • : the host opens door j after you pick door i

13. Monty Hall Problem: Bayes Rule continued • WLOG, i=1, j=3

14. Monty Hall Problem: Bayes Rule continued

15. Monty Hall Problem: Bayes Rule continued • You should switch!

16. Continuous Random Variables • What if X is continuous? • Probability density function (pdf) instead of probability mass function (pmf) • A pdf is any function that describes the probability density in terms of the input variable x.

17. Probability Density Function • Properties of pdf • Actual probability can be obtained by taking the integral of pdf • E.g. the probability of X being between 0 and 1 is

18. Cumulative Distribution Function • Discrete RVs • Continuous RVs

19. Common Distributions • Normal • E.g. the height of the entire population

20. Moments • Mean (Expectation): • Discrete RVs: • Continuous RVs: • Variance: • Discrete RVs: • Continuous RVs:

21. Properties of Moments • Mean • If X and Y are independent, • Variance • If X and Y are independent,

22. Moments of Common Distributions • Uniform • Mean ; variance • Binomial • Mean ; variance • Normal • Mean ; variance

23. Simulating events by Matlab programs Write a program in Matlab to distribute the 52 cards of a deck to 4 people, each getting 13 cards. All the choices must be equally likely. One way to do this is as follows: map each card to a number 1, 2, …, 52. Generate a random permutation of the array a[1 2 … 52], then give the cards a[1:13] to first player, a[14:26] to second player etc.

24. Random permutation generation • We can use ceil(rand()*n) to generate a random number from the set {1, 2, …, n}. • Algorithm generate a random permutation: • 1. Start with array a = [ 1 2 … n] • 2. For j = n: -1: 1 • randomly pick a number r in [1..j]. Switch a[r] and a[j] • 3. Output a.