Probability Theory: Counting in Terms of Proportions

1. Great Theoretical Ideas In Computer Science Probability Theory:Counting in Terms of Proportions Lecture 16 CS 15-251

2. Team A is better than Team B: In any one game, the odds are 6:5 that A will beat B. I.e., A beats B with probability 6/11. What are the odds that A will beat B in the World Series (best of seven games)?

3. equally likely sequences of 100 possible outcomes sequences with half 1’s Let’s start more simply… A fair (unbiased, 50/50) coin is tossed 100 times in a row. What is the probability that you will see half heads and half tails? Probability of seeing such a sequence:

4. Language of Probability A fair coin is tossed 100 times in a row. translates to: Let , the space of all outcomes be the set {H,T}100. Each sequence in  is equally likely, and hence has probability 1/||=1/2100.

5. Language of Probability What is the probability of exactly half heads and half tails? translates to: Let E = {xxhas 50 heads} be the event that we see half heads. Probability that E occurs =

6. Set of sequences with 50H’s and 50T’s Set of all 2100 sequences of {H,T}100. Picture Probability of event E = proportion of E in 

7. What is the probability that if you flip a fair coin 100 times you will see 50 heads in a row? Outcome space  = {H,T}100.Each sequence has probability 1/2100. E = {x   | x has 50 H ‘s in a row} |E| = ?

8. anything HHH 250 50 HHH anything T 249 T HHH 249 T HHH 249 T 249 HHH

9. 23 people are in a room. Suppose that all possible assignments of birthdays to the 23 people are equally likely. What is the probability that two people will have the same birthday?

10. to handle leap years count instead!

11. all sequences in  that have no repeated numbers

12. ½, obviously When we roll a fair die, what is the probability that the result is an even number? True, but let’s take the trouble to say this formally.  = {1,2,3,4,5,6} Each x   is equally likely, i.e., x  , the probability that x occurs is 1/6.

14. table of frequencies (proportions) (probabilities) Suppose that a dice is weighted so that the numbers do not occur with equal frequency.

15. Formalizing probability theory for finite probability spaces: Let  be a finite set called the outcome space or sample space. Any set E is called an event. Any set E={x} for some x  is called an atomic event. A probability distribution on  assigns each event E a real number between 0 and 1 called a probability, proportion, weight, or frequency. More formally,…

16. The probability of an event is the sum of the probabilities of the atomic events that comprise it. Recall: () denotes the power set of  consisting of all subsets of  . A probability distribution or measure P:()  [0,1] is any function satisfying Also: we often abuse notation and write P(x) when we mean P({x}). 1.P() =1

17. Notice that a probability distribution must obey three rules: To develop the notion of probability distribution for infinite , we define probability distributions as those functions satisfying these three rules. • Probabilities are non-negative E P(E)  0 • Probabilities are additive E = A  B, A  B =   P(E) = P(A) + P(B) • The whole space has probability 1. P()=1

18. A probability space is a choice of sample space  and a probability distribution P:()  [0,1] . Quite simple. ={w1, w2, …, wk} is a finite set of elements. wi  , P({wi}) assigns the atomic event {wi} a probability.

19. When P assigns each atomic event in  the same probability (1/||), P is called the uniform distribution. Otherwise, P is said to be a non-uniform distribution.

20. Suppose we roll a red die and a green die. What is the probability that sum is 7 or 11? Two spaces we might consider:

21. We can do it this way because  is uniformly distributed. This way does not care about the distribution. Given that you roll a 7, what is the probability that the red dies shows a 1?

22. The probability of event A given event B is written and is defined to be the proportion of A  B to B  B A

23. What is the probability that the red die shows four given that the green die shows two? A (the event that the red die shows four) B (the event that the green die shows two)

24. One bag has two silver coins, another has two gold coins, and the third has one of each.

25. One of the three bags is selected at random. Then one coin is selected at random from the two in the bag. It turns out to be gold. At last, at last! I’m free from that bag!

26. What is the probability that the other coin in the bag is also gold? 1/2 ?

27. Let G1 be the event that the first coin is gold. Pr[G1] = 1/2 Let G2 be the event that the second coin is gold. Pr[G2 | G1 ] = Pr[G1 and G2] / Pr[G1] = (1/3) / (1/2) = 2/3 Surprised?

28. A and B are independent events if

29. Alternative definition of independent events: events A and B are independent if and only if

30. A1, A2, A3, …, Ak are independent if  subsets of them,

31. Probability spaces: 1. A fair coin toss 2. A bias p coin toss

32. P is uniform E1= event that first coin is heads E2= event that second coin is heads 3. two fair, independent coin tosses

33.  the first coin comes up heads  the second coin comes up heads 4. two bias p independent coin tosses.

34. Thus we conclude:

35. 5. n bias p independent coin tosses x a sequence of nH ’s and T ’s with iH ’s and n-iT ’s E all x  with i heads

36. Team A has probability 6/11 of beating Team B in each individual encounter. What is the probability that A will win the World Series (best of seven) against B? , where i = # of W ’s in x E = {x  |x has at least 4 W ’s }

37. Exercise: Why is it permissible to assume that the two teams play a full seven-game series even if one team wins four games before seven have been played?