N( E) ( ) That is, if the outcomes in sample space S are equally likely, then ( )

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1 Stat 400, secton 2.2 Axoms, Interpretatons and Propertes of Probablty notes by Tm Plachowsk In secton 2., we constructed sample spaces by askng, What could happen? Now, n secton 2.2, we begn askng and answerng the queston, How lkely are the sample ponts and events from our sample space? Concept revew: A sample space S contans all possble outcomes for an experment. The outcomes n a sample space must be mutually exclusve. An event (desgnated wth a captal letter A,, C, etc.) s a subset of the sample space, and wll ncorporate one or more of the outcomes. Example A: ou toss two cons. The sample space s S = { }. a) What are the probabltes for the smple events descrbed n the sample space? A probablty model assgns probabltes to all the events n a sample space. In essence, we wll defne the probablty of an event as the proporton of tmes the event s expected to occur. For smple events that are equally lkely to occur, we can use a unform probablty model, as we dd n Example A-a. Formally, for an event E number of ways E can happen number of smple events n The probablty of an event E = P(E) = = number of possble outcomes number of smple events n N That s, f the outcomes n sample space S are equally lkely, then ( ) ( E) N( E) P E = =. N S N Example A contnued: ou toss two cons. The sample space s S = { HH, HT, TH, TT }, 4 smple events that are equally lkely. A = both cons are heads = probablty that both cons are heads = ( ) E S. = one con s heads and the other s tals = P(one con s heads and the other s tals) = P() =

2 Each event (smple or compound) n a sample space wll have a probablty assocated wth t. These probabltes should satsfy the followng axoms. Axom. For any event E, P(E) 0. Axom 2. For any sample space S, P(S) =. Axom 3. For any nfnte collecton of dsjont (mutually exclusve) events E, E 2, E 3, = ( E E E ) P ( ) P K. 2 3 Snce P ( ) = 0, for a fnte collecton of events we can derve P E = P E = P ( E ) = P ( E ) Example -: ou toss a standard sx-sded de. The sample space s S = {, 2, 3, 4, 5, 6 }, 6 smple events that are equally lkely. A = the number rolled s even = k E k. = the number rolled s at least 2 = P() = C = the number rolled s more than 3 = D = the number rolled s at most 4 = P(D) = Example -2: ou toss two standard sx-sded dce. S = { (, ), (2, ), (3, ), (4, ), (5, ), (6, ), (, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6) } A = at least one of the dce s a =

3 = the sum of the two dce s 9 = P() = C = at least one of the dce s a and the sum of the two dce s 9 = Events A and are mutually exclusve. Example C: ou pck a card from a standard deck of 52 cards. [4 suts: spades (S), hearts (H), damonds (D), clubs (C); 3 cards n each sut: ace (A), kng (K), queen (Q), jack (J), 0, 9, 8, 7, 6, 5, 4, 3, 2 ] S = { A-S, A-H, A-D, A-C, K-S, K-H, K-D, K-C, Q-S, Q-H, Q-D, Q-C,, 2-S, 2-H, 2-D, 2-C }. N = 52. The 52 outcomes are equally lkely. A = the card s an Ace = = the card s a Spade = P() = C = the card s the Ace of Spades = D = the card s an Ace or a Spade = P(D) =

4 So far, n the examples we ve used, t has been possble to lst all of the smple events n a sample space. For larger sample spaces, ths may be overwhelmng, or maybe not even possble. We are gong to borrow somethng from set theory: Venn dagrams. Venn dagrams can be used to keep track of ether the smple events n a sample space or ther assocated probabltes. Example D: For a sample space S = { s, s 2, s 3, s 4, s 5, s 6 } all smple events are equally lkely. Let A = { s, s 2, s 3, s 4 } and let = { s 3, s 4, s 5 }. P() = A = P ( A ) = Venn dagram of smple outcomes N(A) = N() = N ( A ) = N ( A ) = P ( A ) = Venn dagram of probabltes the unon/addton prncple for probablty: P( A ) = P ( A) + P( ) P( A ) Let C = { s 6 }. A C = P ( A C) = C = P ( C) = P ( A C) = P ( C) = = A P ( ) = A. Example C revsted: ou pck a card from a standard deck of 52 cards. [4 suts: spades (S), hearts (H), damonds (D), clubs (C); 3 cards n each sut: ace (A), kng (K), queen (Q), jack (J), 0, 9, 8, 7, 6, 5, 4, 3, 2 ] the complement prncple for probablty: For any event E, P( E ) = P( E) A = the card s an Ace, = the card s a Spade =, P() = P ( A ) = D = the card s an Ace or a Spade, P(D) = E = the card s nether an Ace nor a Spade, P(E) =

5 Example E: Suppose that a box contans 3 blue blocks and 2 yellow blocks. ou pck three blocks wthout replacement. Snce blue and yellow do not have unform probabltes, we cannot smply assgn each a probablty of 0.5. A = the frst block s blue Experment: Pck three blocks wthout replacement. = the frst block s yellow P() = C = f the frst block s blue, the second s also blue D = f the frst block s yellow, the second s also blue P(D) = From the Stat 400 page you can lnk to a supplement, tree dagrams and calculatng probabltes, whch has Example E worked out n some detal, for you to peruse at your lesure. So far, the probabltes encountered have been theoretcal. Probabltes can also be determned n emprcal stuatons, through observatons made about actual phenomena. The number of tmes a gven event occurs n the long run s ts frequency. The proporton of tmes a gven event occurs n the long run s ts relatve frequency. In ths type of stuaton, the probablty of an event E s P(E) = value to whch the relatve frequency stablzes wth an ncreasng number of trals. Example F: A hosptal records the number of days each ICU patent stays n ntensve care. S = {, 2, 3, }. Out of 247 ICU patents n the last 5 years, 536 stayed n ICU two weeks or less. If a patent s selected at random, what s the probablty that she or he wll stay n ICU two weeks or less?

6 Example G: Slver Sprngs, Florda, has a snack bar and a gft shop. The management observes 00 vstors, and counts 65 who eat n the snack bar (F), 55 who make a purchase n the gft shop (G), and 40 who do both. a) What s the probablty that a vstor wll not buy anythng n the gft shop? b) What s the probablty that a vstor wll ether eat n the snack bar or buy somethng n the gft shop? c) What s the probablty that a vstor wll buy somethng n the gft shop but not eat n the snack bar? Example H: In 2009, households were surveyed about health nsurance coverage, wth the followng results. Age 8-64 Age < 8 Publc Plan Prvate Plan Unnsured DATA SOURCE: CDC/NCHS, Natonal Health Intervew Survey, , Famly Core component. Data are based on household ntervews of a sample of the cvlan nonnsttutonalzed populaton. Numbers gven above were extrapolated from summary tables. E = ndvdual s under 8; P(E ) = E 2 = ndvdual has health nsurance; P(E 2 ) = E 3 = ndvdual s under 8 and has health nsurance; P(E 3 ) = E 4 = ndvdual s under 8 or has health nsurance; P(E 4 ) =