STAT 270 - Chapter 3 Probability - People.stat.sfu.ca

STAT 270 - Chapter 3ProbabilityMay 22, 2012Shirin Golchi () STAT270 May 22, 2012 1 / 28

Some definitionsExperiment: Any action that produces dataSample space: The set of all possible outcomes of an experimentDiscrete, e.g.,Continuous, e.g.,Event: A subset of the sample spaceExamples: Flip a coin and roll a die,Shirin Golchi () STAT270 May 22, 2012 2 / 28

Set TheoryVenn diagram: A graphical tool to explain events”A union B” denoted by A ∪ B ≡ A or B:”A intersect B” denoted by A ∩ B ≡ AB ≡ A and B:A compliment denoted by Ā or A c or A ′ :”The empty set” denoted by Ø:”mutually exclusive” or ”disjoint” sets:Shirin Golchi () STAT270 May 22, 2012 3 / 28

de Morgan’s lawsA ∪ B = Ā ∩ ¯BA ∩ B = Ā ∪ ¯BShirin Golchi () STAT270 May 22, 2012 4 / 28

Definitions of probabilityAxiomatic definition (Kolmogrov 1993) P is a probability measure if:Axiom 1. For any event A, P(A) ≥ 0Axiom 2. P(S) = 1Axiom 3. If A 1 , A 2 , . . . are mutually exclusive events thenP(∪A i ) = ∑ P(A i ) (countable additivity)Shirin Golchi () STAT270 May 22, 2012 5 / 28

Useful properties of probabilitiescan be proved from the above three axiomsP(Ā) = 1 − P(A)P(a ∪ B) = P(A) + P(B) − P(A ∩ B)Shirin Golchi () STAT270 May 22, 2012 6 / 28

ExampleSuppose that 55% of all adults regularly consume coffee, 45% regularlyconsume carbonated soda and 70% regularly consume at least one of thesetwo products. What is the probability that a randomly selected adulta. regularly consumes both coffee and soda?b. doesn’t regularly consume any of the products?Shirin Golchi () STAT270 May 22, 2012 7 / 28

useful properties of probabilitiesP(Ø) = 0If A ⊆ B then P(A) ≤ P(B)P(A ∪ B ∪ C) =P(A) + P(B) + P(C) − P(AB) − P(AC) − P(BC) + P(ABC)Shirin Golchi () STAT270 May 22, 2012 8 / 28

Symmetry definition of probabilityApplicable when the experiment has finite number of equally likelyoutcomesP(A) =number of outcomes leading to Atotal number of outcomesExample. Suppose that we flip two fair coins. What is the probability thatwe observe at least one head?Shirin Golchi () STAT270 May 22, 2012 9 / 28

Criticisms to the symmetry definitiondefinition is circular; probability is defined in terms of equally likelyoutcomesRestricts us to finite sample spaces while most interesting probabilityproblems have infinite sample spacesThe definition is mostly useful for calculating probabilities in games ofchance, e.g., dice, cards, coin, etc.Shirin Golchi () STAT270 May 22, 2012 10 / 28

Frequency definitionConsider N identical trialsCriticisms:number of occurances of AP(A) = limN→∞NDoesn’t tell us how to calculate probabilitiesNo mathematical reasons that the limit existsDoes not allow interpretationShirin Golchi () STAT270 May 22, 2012 11 / 28

Conditional probability (important)A and B are events, we are interested in the probability that A occuresgiven that B has occured,P(A|B) =P(A ∩ B)P(B)provided that P(B) ≠ 0.⇒ P(A ∩ B) = P(A|B)P(B)Note that,P(A) = P(A|S)Shirin Golchi () STAT270 May 22, 2012 12 / 28

ExampleA magazine publishes three columns Arts (A), Books (B), and Cinema(C). The probability that a randomly selected reader is interested inArts is 0.14Books is 0.23Cinema is 0.37Arts and Books is 0.08Arts and Cinema is 0.09Books and Cinema is 0.13Arts, Books, and Cinema is 0.05Calculate the probabilities that a randomly selected reader is interested ina. Arts given that s/he is interested in Booksb. Cinema given that s/he is interested in at least one of Arts or Booksc. at least one of Cinema or Books given that s/he is interested in Artsd. Books given that s/he is interested in at least one of the columnsShirin Golchi () STAT270 May 22, 2012 13 / 28

The law of the total probabilityConsider events A and B 1 , B 2 , . . . where B i s form a partition of S, i.e.,they are disjoint and S = ∪ ∞ i=1 B i, thenP(A) = P(∪ ∞ i=1AB i ) =∞∑P(AB i ) =i=1∞∑P(A|B i )P(B i )i=1Shirin Golchi () STAT270 May 22, 2012 14 / 28

IndependenceThe events A and B are independent iff (if and only if)P(A ∩ B) = P(A)P(B)meaning that the occurance of either of them does not affect theprobability of the other.Shirin Golchi () STAT270 May 22, 2012 15 / 28

ExampleToss afair coin and roll a die. Define,A: the event that we observe a head on the coinB: the event that we observe a 6 on the dieAre A and B independent?Shirin Golchi () STAT270 May 22, 2012 16 / 28

Tipprobability of A if or given that . . . has occured ≡ P(A| . . .)Shirin Golchi () STAT270 May 22, 2012 17 / 28

Birthday problemIf there are n people in a room, what is the probability that at least two ofthem were born in the same day of the year?Shirin Golchi () STAT270 May 22, 2012 18 / 28

IndependenceProposition. Suppose P(B) ≠ 0, then A and B are independent iffP(A|B) = P(A).Proof.Definition. A 1 , A 2 , . . . , A k are mutually independent ifffor m = 2, 3, . . . , k.P(A 1 ∩ A 2 ∩ . . . ∩ A m ) = P(A 1 )P(A 2 ) . . . P(A m )Shirin Golchi () STAT270 May 22, 2012 19 / 28

Combinatorial rulesThe number of permutations (arrangements) of n distinct objects isn! = n(n − 1) . . . 1, 0! = 1.e.g.The number of permutations of r objects chosen from n distinctobjects is n (r) = n!(n−r)!, ”n to r factors”e.g.The number of combinations of r objects chosen from n distinctobjects (order is not important)(n choose r)( ) n= n(r) n!=r r! (n − r)!r!e.g.Shirin Golchi () STAT270 May 22, 2012 20 / 28

Combinatorial rulesProposition. The number of ways of partitioning n distinct objects intotwo groups of sizes r and n − r is ( )nrCorollary. ( ) n=r( n)n − rProposition The number of ways of partitioning n distinct objects into kgroups of sizes n 1 , . . . , n k where n = n 1 + . . . + n k isproof:n!n 1 ! . . . n k !Shirin Golchi () STAT270 May 22, 2012 21 / 28

ExampleExample 3.10. (text) 100 students, 20 female. What is the probabilitythat in a randomly drawn sample of size five at least two are female?Shirin Golchi () STAT270 May 22, 2012 22 / 28

ExampleAn academic faculty with 5 faculty members narrowed its choice fordepartment head to either candidate A or B. Each member then voted ona slip of paper. Suppose there are 3 votes for A and 2 for B. If the slipsare selected for tallying in random order, what is the probability that Aremains ahead of B?Shirin Golchi () STAT270 May 22, 2012 23 / 28

ExampleExample 3.12. (text) Roll a die; if 6 is obtained draw a ball from box Acontaining 3 white balls and 2 black balls, if 1, . . . , 5 is obtained draw aball from box B containing 2 white balls and 4 black balls.a. What is the probability of obtaining a white ball?b. If the chosen ball is white what is the probability that it is chosen frombox A?Shirin Golchi () STAT270 May 22, 2012 24 / 28

Playing cardsSome information,Ordinary deck:52 cards13 denominations: ace, 2, 3,. . . , 10, jack, queen, king4 suits: diamond ♦, heart ♥, club ♣, spade ♠A little poker lesson:Shirin Golchi () STAT270 May 22, 2012 25 / 28

ExampleExample 3.13. (text) In a hand of five dealt from an ordinary deck whatis the probability ofa. three of a kind?b. two pair?c. a straight flush?Shirin Golchi () STAT270 May 22, 2012 26 / 28

ExampleExample 3.14. (text) 649 lottery: 6 balls are chosen from an urncontaining 49 balls numbered from 1-49. Particioants buy tickets andselect 6 different numbers between 1 and 49. Calculate the probability of,a. winning the jackpot (all numbers match)b. two of the numbers match.Shirin Golchi () STAT270 May 22, 2012 27 / 28

ExampleExample 3.17 (text) 3 marbles are drawn from a bag containing 4 redmarbles and 6 black marbles. What is the probability that all threemarbles are red when they are drawna. with eplacement?b. without replacement?repeat the example with the bag containing 40 red marbles and 60 blackmarbles.Shirin Golchi () STAT270 May 22, 2012 28 / 28