IndependenceLecture 301/10/2013(Lecture 3) 01/10/2013 1 / 18Probability rules1. Conditional probability of A given BP(A|B) =P(A and B)P(B)Conditional probability of B given AP(B|A) =P(A and B)P(A).2. Multiplication RuleP(A and B) = P(B) · P(A|B).P(A and B) = P(A) · P(B|A).(Lecture 3) 01/10/2013 2 / 18Probability rules3. General Addition Rule (the General OR Rule)P(A or B) = P(A) + P(B) − P(A and B).4. Total ProbabilityP(A) = P(A and B) + P(A and Bc)5. Complement RuleP(A) = 1 − P(Ac)P(A|B) = 1 − P(Ac|B)(Lecture 3) 01/10/2013 3 / 18ExampleStandard Deck of cards- A standard deck of cards has 52 cards.- 4 suits (Hearts , Spades , Diamonds , Clubs )- Each suit has:INumbers from 1-10IFaces (King , Queen , Jack )We draw a card at random.What is the probability that we draw a King or a Heart?(Lecture 3) 01/10/2013 4 / 18ExampleUse the general addition rule:P(King) =452P(Hearts) =1352P(King and Hearts) = P(King of Hearts) =152P(King or Hearts) = P(King) + P(Hearts) − P(King and Hearts)=452+1352−152=1652=413(Lecture 3) 01/10/2013 5 / 18ExampleRoll a fair die.What is the probability to roll an even number or a 3?P(Roll Even) = P({2,4,6}) =36=12P(Roll 3) = P({3}) =16- Rolling an {Even number} and {Rolling a 3} are two mutuallyexclusive events.P(Even or 3) = P(Roll Even) + P(Roll 3)=16+36=46=23(Lecture 3) 01/10/2013 6 / 18IndependenceA and B are independent events ifP(B|A) = P(B) or P(A|B) = P(A).Note: If A and B are independent events then (i) Acand B, (ii) A andBc, (iii) Bcand A, and (iv) Acand Bcare also independent events, i.e.,P(Ac|B) = P(Ac), P(A|Bc) = P(A),P(Bc|A) = P(Bc), P(Ac|Bc) = P(Ac).Joint Probability of two Independent events:P(A and B) = P(A) · P(B|A) = P(B) · P(A|B) (Multiplication Rule)= P(A) · P(B) (Independence).(Lecture 3) 01/10/2013 7 / 18Is King independent of Hearts?P(King) =452P(Hearts) =1352P(King and Hearts) = P(King of Hearts) =152P(King) · P(Hearts) =452·1352=52522=152They are independent!(Lecture 3) 01/10/2013 8 / 18Is King independent of Hearts?Another way to check for independence:P(King) =452=113P(King|Hearts) =P(King and Hearts)P(Hearts)=P(King of Hearts)P(Hearts)=1521352=113P(King|Hearts) = P(King)They are independent!(Lecture 3) 01/10/2013 9 / 18Is Face independent of King?P(King) =452P(Face) =1252P(King and Face) = P(King) =452P(King) · P(Face) =452·1252=48522They are NOT independent!(Lecture 3) 01/10/2013 10 / 18Is Face independent of King?Another way to check for independence:P(King) =452=113P(King|Face) =P(King and Face)P(Face)=P(King)P(Face)=4521252=412=13P(King|Face) 6= P(King)They are NOT independent!(Lecture 3) 01/10/2013 11 / 18Mutually Exclusive vs. Independent EventsIndependent Events Mutually Exclusive EventsP(A and B) = P(A) · P(B) P(A and B) = 0P(A|B) = P(A) P(A|B) = 0P(A or B) = P(A) + P(B) − P(A) · P(B) P(A or B) = P(A) + P(B)(Lecture 3) 01/10/2013 12 / 18Rules of ThumbNot −→ 1 – ProbabilityOR & Mutually Exclusive −→ AddAND & Independence −→ Multiply(Lecture 3) 01/10/2013 13 / 18Successive EventsP(Heads then Tails) =1212=14P(Roll ) =1616=136P(Roll 1 and then an even number) =1612=112(Lecture 3) 01/10/2013 14 / 18Repeated EventsA = Flip Heads 6 times (HHHHHH)P(A) =12×12×12×12×12×12=126=164B = Get HTHTTHP(B) =164(Lecture 3) 01/10/2013 15 / 18Sampling without ReplacementA jar contains 6 blue and 6 red marbles.You pull out 3.P(1st is blue) =612= 0.5P(2nd is blue | 1st is blue) =511= 0.45P(3rd is blue|1st two were blue) =410= 0.4P(1st two are blue) = P(1st is blue) · P(2nd is blue | 1st is blue)= (0.5)(0.45) = 0.225(Lecture 3) 01/10/2013 16 / 18Sampling without ReplacementDraw two cards from a standard deck of cards:A = { First card is a King }B = { Second card is a King }P(B|A) =351P(Draw two Kings) = P(A and B)= P(A) · P(B|A)=452·351= 0.0045(Lecture 3) 01/10/2013 17 / 18See you on Monday!(Lecture 3) 01/10/2013 18 /