1. An Introduction to Probability Probability, odds, outcomes and events

2. Definitions 5 Experiment – any observation or measurement of a random phenomenon (the outcome cannot be predicted with certainty) Simple event – the most basic outcome of an experiment Outcomes– possible results of an experiment Sample Space – the set of all possible outcomes (i.e., the collection of all simple events Theoretical Probability: n(E)/n(S), ( # of favorable outcomes)/(total # of outcomes)

3. Terminology 3 Empirical probability: P(E) = (# of times event E occurred)/(# of times experiment was performed) –arrived at by experimentation Did: P(face card) = 12/52=0.2307…. P(not a face card)=1-0.2307=0.7693… =40/52 Law of Large Numbers (Law of averages): An an experiment is repeated more and more times, the proportion of outcomes favorable to any particular event will come closer and closer to the theoretical probability of that event

4. Probability questions 5 “I have two children. One is a boy, and one is as a…..” What is the chance that I have two boys? Ans: 1/3: outcomes-- BB, BG, GG “I have two children. The older is a boy, and …..” What is the chance that I have two boys? Ans: ½ Experiments rolling die: rollDie.java; compute probabilityrollDie.java

5. Birthday problem: 4 How many people are needed in a room so that the probability that there are two people whose birthdays are the exactly the same day is roughly ½? How many pairs of dates? 365x365; How many pairs which are guaranteed that no two the same: 365x364; (364*365)/(365*365) = 364/365=0.9972… If probability no two the same is 0.9972, then P(2 same)= 1-0.9972=0.00273…

6. Birthday problem: (cont.) 3 3, no two the same: (365*364*363)/(365*365*365)=0.9917; P(same)=1-0.9917…= 0.0082… 4 w/ two the same 1- (365*364*363*362)/(365)^4 = 0.01635; Pattern:

7. Birthday problem: (cont.) # people in rmP(2 sharing birthday) 50.027 100.116 200.252 250.411 500.970 700.994 800.99991 900.999993

8. Birthday problem: (cont.) 2 Thus, at 23 people, the P(2 people share birthday) = 0.5072 Class experiment w/ birthdays

9. Computing probabilities 3 P flush, full house; P red or face card? Need general addition Rule: P(A or B)=P(A)+P(B)-P(A  B) P(R or F)= P(R)+P(F)-P(R  F) =(26+12-6)/52=8/13 Them: club or 2, spade or diamond

10. Conditional probabilities 3 P(B|A)=P(A  B)/P(A), where P(B|A) is the probability of event B, given that A has already happened Ex/2 fair coins are tossed –at least one is a head -- find the probability that both are heads: A= 1 head; B = both heads; P(B|A)= P(A  B)/P(A)=1/4/3/4=1/3

11. Conditional probabilities 6 P(B|A)=P(A  B)/P(A), where P(B|A) is the probability of event B, given that A has already happened Ex/2 fair coins are tossed –at least one is a head -- find the probability that both are heads: A= 1 head; B = both heads; P(B|A)= P(A  B)/P(A)=1/4//3/4=1/3 Do ex 36,44 on sheet Hw/ 1,3,37,45 Hi/ 2,35,38,43,46