Intro to Probability

Transcription

1 Intro to Probability February 26, 2019 Data Science CSCI 1951A Brown University Instructor: Ellie Pavlick HTAs: Wennie Zhang, Maulik Dang, Gurnaaz Kaur Content stolen largely from Dan Potter s 2016 version of this course, which were in turn stolen largely from the version of MIT s course (18.05)

2 Announcements Projects: You ll be receiving feedback on final project proposals soon. Mentor TA will reach out to schedule an initial check in sometime between March 2 - March 8 Comments about project scope and such MR homework its okay to change the number of mappers/reducers as long as the output is correct 2

3 Today Probability Spaces, Probability Functions, Events Bayesian Statistics Random Variables 3

4 Today Probability Spaces, Probability Functions, Events Bayesian Statistics Random Variables A bit of a laundry list, sorry. 4

5 Today Probability Spaces, Probability Functions, Events Bayesian Statistics Random Variables 5

6 Statistics vs. Prob. Theory Probability theory: mathematical theory that describes uncertainty. Distributions/parameters are known Statistics: techniques for extracting useful information from data. Distributions/parameters are generally unknown and need to be estimated from the data 6

7 Statistics vs. Prob. Theory Probability theory: mathematical theory that describes uncertainty. Distributions/parameters are known Statistics: techniques for extracting useful information from data. Distributions/parameters are generally unknown and need to be estimated from the data 7

8 Statistics vs. Prob. Theory Probability theory: mathematical theory that describes uncertainty. Distributions/parameters are known Statistics: techniques for extracting useful information from data. Distributions/parameters are generally unknown and need to be estimated from the data 8

9 Statistics vs. Prob. Theory Probability theory: mathematical theory that describes uncertainty. Distributions/parameters are known Statistics: techniques for extracting useful information from data. Distributions/parameters are generally unknown and need to be estimated from the data 9

10 Statistics vs. Prob. Theory Probability theory: mathematical theory that describes uncertainty. Distributions/parameters are known Statistics: techniques for extracting useful information from data. Distributions/parameters are generally unknown and need to be estimated from the data 10

11 The bigger picture Start with real world phenomenon/observations Make assumptions about the underlying model Fit the parameters of the model based on data 11

12 The bigger picture Whether a coin is heads or tails What a person will say next Whether someone will click on an add Start with real world phenomenon/observations Make assumptions about the underlying model Fit the parameters of the model based on data 12

13 The bigger picture You *always* make assumptions about the structure of the process that is generating the data Start with real world phenomenon/observations Make assumptions about the underlying model Fit the parameters of the model based on data All models are bad, but some are useful 13

14 The bigger picture Goal is always to explain the data Start with real world phenomenon/observations Make assumptions about the underlying model Fit the parameters of the model based on data Two typical (different) use cases: 1) understand the underlying model better 2) make predictions 14

15 Probability Space h,f,pi 15

16 Probability Space h,f,pi the set of all possible outcomes of the random process modeled 16

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sha1_base64="nxqvnkzkpjxsq1zasljwet1ynkg=">aaaccxicbvdlssnafj34rpuvdelmsah1uxirdcmurxazwt6gcweynbrdj5mwd6gebt34k25ckolwp3dn3zhps9dwawpnnnmvd+6jmkalcpxva2l5zxvtvbjr3dza3tm19/bbmtuckxzowsq6ezkeuu5aiipgupkgkiky6usj68lvpbahacrv1tgjqyignmyui2wk0ize3y/oaossppampcfwevpsj6by6kud2jwn4uwbf4lbkhoo4yx2l99psu4iv5ghkxuuk6kgr0jrzmik6mtjmorhaeb6hnkuebnk00sm8ngofrinwjyu4ft9pzgjrmpxepnobkmhnpck8t+vp1v8eesuz1orjmelys2gsmerc+xtqbbiy0mqftt8feiheggre17vhodon7xi2qcn12m4d2e15luzrwucginqby44b01wczzqahg8gmfwct6sj+vferc+zq1lvjlzap7a+vwbou6ygw==</latexit> <latexit sha1_base64="nxqvnkzkpjxsq1zasljwet1ynkg=">aaaccxicbvdlssnafj34rpuvdelmsah1uxirdcmurxazwt6gcweynbrdj5mwd6gebt34k25ckolwp3dn3zhps9dwawpnnnmvd+6jmkalcpxva2l5zxvtvbjr3dza3tm19/bbmtuckxzowsq6ezkeuu5aiipgupkgkiky6usj68lvpbahacrv1tgjqyignmyui2wk0ize3y/oaossppampcfwevpsj6by6kud2jwn4uwbf4lbkhoo4yx2l99psu4iv5ghkxuuk6kgr0jrzmik6mtjmorhaeb6hnkuebnk00sm8ngofrinwjyu4ft9pzgjrmpxepnobkmhnpck8t+vp1v8eesuz1orjmelys2gsmerc+xtqbbiy0mqftt8feiheggre17vhodon7xi2qcn12m4d2e15luzrwucginqby44b01wczzqahg8gmfwct6sj+vferc+zq1lvjlzap7a+vwbou6ygw==</latexit> <latexit sha1_base64="5wijlkmavxhls+qkbmwufybezgq=">aaab/xicbvdlssnafl3xwesrpnzubovgqiqikk6kgrisyh/qhdkzttqhk0mymqg1fh/fjqtf3pof7vwbj20w2npg4hdovcy5j0g4u9pxvq2fxaxlldxswnl9y3nr297zbao4lyq2smxj2q6wopwj2tbmc9pojmvrwgkrgf7lfuubssvica9hcfuj3bcszarri3xt/fofukaejpexyt0iguxujlp2xak6e6b54hakagxqxfvl68ukjajqhgoloq6tad/dujpc6bjspyommaxxn3ymftiiys8m6cfoycg9fmbspkhrrp29kefiqveummk8opr1cve/r5pq8nzpmehstqwzfhsmhjlj8ypqj0lknb8zgolkjisiaywx0aawsinbnt15njrpqq5tdw9pk7xloo4shmahhimlz1cdg6hdawg8wjo8wpv1zl1y79bhdhtbknb24a+szx+ws5qw</latexit> <latexit sha1_base64="5wijlkmavxhls+qkbmwufybezgq=">aaab/xicbvdlssnafl3xwesrpnzubovgqiqikk6kgrisyh/qhdkzttqhk0mymqg1fh/fjqtf3pof7vwbj20w2npg4hdovcy5j0g4u9pxvq2fxaxlldxswnl9y3nr297zbao4lyq2smxj2q6wopwj2tbmc9pojmvrwgkrgf7lfuubssvica9hcfuj3bcszarri3xt/fofukaejpexyt0iguxujlp2xak6e6b54hakagxqxfvl68ukjajqhgoloq6tad/dujpc6bjspyommaxxn3ymftiiys8m6cfoycg9fmbspkhrrp29kefiqveummk8opr1cve/r5pq8nzpmehstqwzfhsmhjlj8ypqj0lknb8zgolkjisiaywx0aawsinbnt15njrpqq5tdw9pk7xloo4shmahhimlz1cdg6hdawg8wjo8wpv1zl1y79bhdhtbknb24a+szx+ws5qw</latexit> <latexit sha1_base64="5wijlkmavxhls+qkbmwufybezgq=">aaab/xicbvdlssnafl3xwesrpnzubovgqiqikk6kgrisyh/qhdkzttqhk0mymqg1fh/fjqtf3pof7vwbj20w2npg4hdovcy5j0g4u9pxvq2fxaxlldxswnl9y3nr297zbao4lyq2smxj2q6wopwj2tbmc9pojmvrwgkrgf7lfuubssvica9hcfuj3bcszarri3xt/fofukaejpexyt0iguxujlp2xak6e6b54hakagxqxfvl68ukjajqhgoloq6tad/dujpc6bjspyommaxxn3ymftiiys8m6cfoycg9fmbspkhrrp29kefiqveummk8opr1cve/r5pq8nzpmehstqwzfhsmhjlj8ypqj0lknb8zgolkjisiaywx0aawsinbnt15njrpqq5tdw9pk7xloo4shmahhimlz1cdg6hdawg8wjo8wpv1zl1y79bhdhtbknb24a+szx+ws5qw</latexit> <latexit sha1_base64="5wijlkmavxhls+qkbmwufybezgq=">aaab/xicbvdlssnafl3xwesrpnzubovgqiqikk6kgrisyh/qhdkzttqhk0mymqg1fh/fjqtf3pof7vwbj20w2npg4hdovcy5j0g4u9pxvq2fxaxlldxswnl9y3nr297zbao4lyq2smxj2q6wopwj2tbmc9pojmvrwgkrgf7lfuubssvica9hcfuj3bcszarri3xt/fofukaejpexyt0iguxujlp2xak6e6b54hakagxqxfvl68ukjajqhgoloq6tad/dujpc6bjspyommaxxn3ymftiiys8m6cfoycg9fmbspkhrrp29kefiqveummk8opr1cve/r5pq8nzpmehstqwzfhsmhjlj8ypqj0lknb8zgolkjisiaywx0aawsinbnt15njrpqq5tdw9pk7xloo4shmahhimlz1cdg6hdawg8wjo8wpv1zl1y79bhdhtbknb24a+szx+ws5qw</latexit> Probability Space Valid Probability Function: 0 apple P (E) apple 1 8E 2 F h,f,pi P ( ) =1 P ( [ i E i )= X i P (E i ) Probability function which assigns a real number to each event in F P : F! R 20

21 Over-used Example Tossing a fair coin once h,f,pi 21

22 Over-used Example Tossing a fair coin once h,f,pi {H, T} 22

23 <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> Over-used Example Tossing a fair coin once h,f,pi {H, T} F =2 23

24 <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> Over-used Example Tossing a fair coin once h,f,pi {H, T} F =2 {}, {H}, {T}, {H, T} 24

25 <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> {} -> 0 Over-used Example {H} -> 0.5 {T} -> 0.5 {H, T} ->??? Tossing a fair coin once h,f,pi {H, T} F =2 {}, {H}, {T}, {H, T} 25

26 <latexit sha1_base64="sfqrxgg/yvfglqxtxrjphqwpqz0=">aaab83icbvbnswmxem3wr1q/qh69bivgqewwqs9curbvvraf0f1lnp1tq5pskmsfsvrvepggiff/jdf/jwm7b219mpb4b4azewhcmtau++0uvlbx1jekm6wt7z3dvfl+quvhqalqpdgpvsckgjit0dtmcogkcogiobtd0fxubz+b0iywd2acqcdiqlkiuwks5n/gs1x79o8edeivxhgr7gx4mxg5qaacjv75y+/hnbugdeve667njibiidkmcpiu/frdquiidkbrqsqcdjdnbp7ge6v0crqrw9lgmfp7iinc67eibacgzqgxvan4n9dntxqrzewmqqfj54uilgmt42kaum8uumphlhcqml0v0yfrhbobu8mg4c2+vexatarnvr37s0r9ko+jii7qmtpfhjphdxslgqijkerqm3pfb07qvdjvzse8tedkm4fod5zph0nvkio=</latexit> <latexit 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Tossing a fair coin once h,f,pi {H, T} F =2 Valid Probability Function: 0 apple P (E) apple 1 8E 2 F P ( ) =1 {}, {H}, {T}, {H, T} P ( [ i E i )= X i P (E i ) 26

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28 Another example Language modeling: predict what a person will say. h,f,pi 28

29 Another example Language modeling: predict what a person will say. h,f,pi English vocabulary 29

30 Another example Language modeling: predict what a person will say. h,f,pi English vocabulary All sequences of words 30

31 Another example Language modeling: predict what a person will say. h,f,pi English vocabulary All sequences of words P(w) for every w 31

32 Events Experiment: a repeatable procedure Sample space: set of all possible outcomes Ω Event: a subset of the sample space Discrete listable/countable; can be infinite (e.g. {1, 2, 3, 4, }) or finite (e.g. {a, b, z}) Continuous not discrete :) 32

33 Events Experiment: a repeatable procedure Sample space: set of all possible outcomes Ω Event: a subset of the sample space Discrete listable/countable; can be infinite (e.g. {1, 2, 3, 4, }) or finite (e.g. {a, b, z}) Continuous not discrete :) 33

34 Events Experiment: a repeatable procedure Sample space: set of all possible outcomes Ω Event: a subset of the sample space Discrete listable/countable; can be infinite (e.g. {1, 2, 3, 4, }) or finite (e.g. {a, b, z}) Continuous not discrete :) 34

35 Events Experiment: a repeatable procedure Sample space: set of all possible outcomes Ω Event: a subset of the sample space Discrete listable/countable; can be infinite (e.g. {1, 2, 3, 4, }) or finite (e.g. {a, b, z}) Continuous not discrete :) 35

36 Events Experiment: a repeatable procedure Sample space: set of all possible outcomes Ω Event: a subset of the sample space Discrete listable/countable; can be infinite (e.g. {1, 2, 3, 4, }) or finite (e.g. {a, b, z}) Continuous not discrete :) 36

37 Events Experiment: a repeatable procedure Sample space: set of all possible outcomes Ω Event: a subset of the sample space Discrete listable/countable; can be infinite (e.g. {1, 2, 3, 4, }) or finite (e.g. {a, b, z}) Continuous not discrete :) 37

38 Events Experiment: Flip a coin once Both heads {TT} {HH} Both tails Disjoint Events 38

39 Events Experiment: Flip a coin once At least one head {HT} {TH} {TT} {HH} Both tails Disjoint Events 39

40 Events Experiment: Flip a coin once At least one head {HT} {TH} {TT} {HH} Both tails Partition 40

41 <latexit sha1_base64="z/eoh2ad8rquddcxasc00obp3jm=">aaacehicbvdlsgmxfl1tx7w+rl26craxrswziuhgqhxjsoj9qduutjppqzmpkoxqhn6cg3/fjqtf3lp059+yawehrqdczj3nxpj73igzqszr28gtla+sruxxcxubw9s75u5eu4axilrbqh6ktosl5sygdcuup+1iuoy7nlbc0u3qtx6okcwm7tu4oo6pbwhzgmfksz3zuf66rl0sr6hwrldiv2v0oi9dnakzh1ovzxatijufwir2roqqod4zv7r9kmq+drthwmqobuxksbbqjha6kxrjssnmrnhao5og2kfssayltdcrvvric4u+gujt9fdegn0px76ro32shnles8x/ve6svesnyueukxqq2unezjekuzoo6jnbiejjttartp8vksewmcidyughym+vveiazxxbqth358vqlysjdwdwccww4qkqcat1aacbr3igv3gznowx4934mlxmjgxmh/7a+pwbn3cxrw==</latexit> <latexit sha1_base64="z/eoh2ad8rquddcxasc00obp3jm=">aaacehicbvdlsgmxfl1tx7w+rl26craxrswziuhgqhxjsoj9qduutjppqzmpkoxqhn6cg3/fjqtf3lp059+yawehrqdczj3nxpj73igzqszr28gtla+sruxxcxubw9s75u5eu4axilrbqh6ktosl5sygdcuup+1iuoy7nlbc0u3qtx6okcwm7tu4oo6pbwhzgmfksz3zuf66rl0sr6hwrldiv2v0oi9dnakzh1ovzxatijufwir2roqqod4zv7r9kmq+drthwmqobuxksbbqjha6kxrjssnmrnhao5og2kfssayltdcrvvric4u+gujt9fdegn0px76ro32shnles8x/ve6svesnyueukxqq2unezjekuzoo6jnbiejjttartp8vksewmcidyughym+vveiazxxbqth358vqlysjdwdwccww4qkqcat1aacbr3igv3gznowx4934mlxmjgxmh/7a+pwbn3cxrw==</latexit> <latexit sha1_base64="z/eoh2ad8rquddcxasc00obp3jm=">aaacehicbvdlsgmxfl1tx7w+rl26craxrswziuhgqhxjsoj9qduutjppqzmpkoxqhn6cg3/fjqtf3lp059+yawehrqdczj3nxpj73igzqszr28gtla+sruxxcxubw9s75u5eu4axilrbqh6ktosl5sygdcuup+1iuoy7nlbc0u3qtx6okcwm7tu4oo6pbwhzgmfksz3zuf66rl0sr6hwrldiv2v0oi9dnakzh1ovzxatijufwir2roqqod4zv7r9kmq+drthwmqobuxksbbqjha6kxrjssnmrnhao5og2kfssayltdcrvvric4u+gujt9fdegn0px76ro32shnles8x/ve6svesnyueukxqq2unezjekuzoo6jnbiejjttartp8vksewmcidyughym+vveiazxxbqth358vqlysjdwdwccww4qkqcat1aacbr3igv3gznowx4934mlxmjgxmh/7a+pwbn3cxrw==</latexit> Events Experiment: Flip a coin once At least one head {TH} {HH} {HT} {TT} Second flip is tails Inclusion-Exclusion Priciple P (A [ B) =P (A)+P (B) P (A \ B) <latexit sha1_base64="z/eoh2ad8rquddcxasc00obp3jm=">aaacehicbvdlsgmxfl1tx7w+rl26craxrswziuhgqhxjsoj9qduutjppqzmpkoxqhn6cg3/fjqtf3lp059+yawehrqdczj3nxpj73igzqszr28gtla+sruxxcxubw9s75u5eu4axilrbqh6ktosl5sygdcuup+1iuoy7nlbc0u3qtx6okcwm7tu4oo6pbwhzgmfksz3zuf66rl0sr6hwrldiv2v0oi9dnakzh1ovzxatijufwir2roqqod4zv7r9kmq+drthwmqobuxksbbqjha6kxrjssnmrnhao5og2kfssayltdcrvvric4u+gujt9fdegn0px76ro32shnles8x/ve6svesnyueukxqq2unezjekuzoo6jnbiejjttartp8vksewmcidyughym+vveiazxxbqth358vqlysjdwdwccww4qkqcat1aacbr3igv3gznowx4934mlxmjgxmh/7a+pwbn3cxrw==</latexit> 41

42 Clicker Question! 42

43 Clicker Question! Experiment: toss a coin 3 times. Which of following equals the event exactly two heads? (a) (b) (c) {THH,HTH,HHT,HHH} {HHT} {THH, HTH, HHT} 43

44 Clicker Question! Experiment: toss a coin 3 times. Which of following equals the event exactly two heads? (a) (b) (c) {THH,HTH,HHT,HHH} {HHT} {THH, HTH, HHT} 44

45 Clicker Question! We have a class of 50 students. 25 are male and 20 have brown eyes. If we randomly select a student, what bounds can we put on the probability that they are male or have brown eyes? (a) less than 0.4 (b) between 0.4 and 0.5 (c) between 0.4 and 0.9 (d) between 0.5 and 0.9 (e) greater than

46 Clicker Question! We have a class of 50 students. 25 are male and 20 have brown eyes. If we randomly select a student, what bounds can we put on the probability that they are male or have brown eyes? (a) less than 0.4 (b) between 0.4 and 0.5 (c) between 0.4 and 0.9 (d) between 0.5 and 0.9 (e) greater than

47 Clicker Question! We have a class of 50 students. 25 are male and 20 have brown eyes. If we randomly select a student, what bounds can we put on the probability that they are male or have brown eyes? Male Brown Eyes Neither p = 45/50 = 90% 47

48 Clicker Question! We have a class of 50 students. 25 are male and 20 have brown eyes. If we randomly select a student, what bounds can we put on the probability that they are male or have brown eyes? Male Brown Eyes Neither p = 25/50 = 50% 48

49 <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> Conditional Probability P (A B) = P (A \ B) P (B) 49

50 <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> Conditional Probability P (A B) = P (A \ B) P (B) At least one head {TH} {HH} 50 {HT} {TT} Second flip is tails

51 <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> <latexit sha1_base64="rrhy1x47qs2l86xm0qm/tmwenjs=">aaaccnicbvc7tsmwfl3hwcorwmhiqjdapuoqeixipsymraipqakqx3vaq44t2q5sftqz8cssdcdeyhew8te4bqzoozktc8+5v/y9fsyz0o7zbs0tr6yurec28ptb2zu79t5+q0wjjlroih7jlo8v5uzqumaa01yskq59tpv+8hrin++pvcwsd3ou006i+4ifjgbtpk59vctepvrl6bj5gcqknsxyci5rttq2hbm7dsepo1ogrejmpaazal37y+tfjamp0irjpdque+toiqvmhnnx3ksujtez4j5tgypwsfunna4yridg6aegkuyijabq74kuh0qnqt90hlgp1lw3ef/z2okoljope3giqsczh4keix2hss6oxyqlmo8mwuqy81debtheok16eroco7/yimmcll2n7n6efsrvli4chmixfmgfc6jaddsgdgqe4rle4c16sl6sd+tj1rpkztmh8afw5w/wspfr</latexit> Conditional Probability P (A B) = P (A \ B) P (B) At least one head {TH} {HH} 51 {HT} {TT} Second flip is tails

52 Clicker Question! 52

53 Clicker Question! Toss a coin 4 times. Let A = at least three heads. Let B = first toss is tails What is P(A B)? (a) 1/16 (b) 1/8 (c) 1/4 (d) 1/5 53

54 Clicker Question! Toss a coin 4 times. Let A = at least three heads. Let B = first toss is tails What is P(A B)? (a) 1/16 (b) 1/8 (c) 1/4 (d) 1/5 54

55 Clicker Question! Toss a coin 4 times. Let A = at least three heads. Let B = first toss is tails What is P(A B)? (a) 1/16 (b) 1/8 (c) 1/4 (d) 1/5 HHHH HHHT HTTH THTT THHH TTHH HTHT TTHT HTHH THTH HHTT TTTH HHTH THHT 55 HTTT TTTT

56 Clicker Question! Toss a coin 4 times. Let A = at least three heads. Let B = first toss is tails What is P(A B)? (a) 1/16 (b) 1/8 (c) 1/4 (d) 1/5 HHHH HHHT HTTH THTT THHH TTHH HTHT TTHT HTHH THTH HHTT TTTH HHTH THHT 56 HTTT TTTT

57 Clicker Question! Toss a coin 4 times. Let A = at least three heads. Let B = first toss is tails What is P(B A)? (a) 1/16 (b) 1/8 (c) 1/4 (d) 1/5 HHHH HHHT HTTH THTT THHH TTHH HTHT TTHT HTHH THTH HHTT TTTH HHTH THHT 57 HTTT TTTT

58 Clicker Question! Toss a coin 4 times. Let A = at least three heads. Let B = first toss is tails What is P(B A)? (a) 1/16 (b) 1/8 (c) 1/4 (d) 1/5 HHHH HHHT HTTH THTT THHH TTHH HTHT TTHT HTHH THTH HHTT TTTH HHTH THHT 58 HTTT TTTT

59 <latexit sha1_base64="lmy2vbh0rfx0ghrpjulcdyldrmo=">aaab/3icbvdlsgmxfl1tx7w+rgu3bojfaddlrgtdclvuxi5gh9aojznm2tbmzkgyqqld+ctuxcji1t9w59+ytrpq1gmxts65l9x7goqzpr3n28qtrk6tb+q3c1vbo7t79v5bq8wpjlroyh7lvoav5uzqumaa01yiky4ctpvb8gbqnx+ovcww93quud/cfcfcrra2utc+8krxqenwgmpldixmq+yvauwuxxqqzgxombgzkuigr2t/dxoxssmqnofyqbbrjnofy6kz4xrs6kskjpgmcz+2dru4osofz/afofoj9fays1nco5n6e2kmi6vguwa6i6whatgbiv957vshl/6yistvvjd5r2hkky7rnazuy5iszuegyckz2rwrazayabnzwytglp68tbpnfdepuhfnxwotiympx3acjxdhaqpwcx7ugcajpmmrvflp1ov1bn3mw3nwnnmif2b9/gdsbzk9</latexit> <latexit 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sha1_base64="lmy2vbh0rfx0ghrpjulcdyldrmo=">aaab/3icbvdlsgmxfl1tx7w+rgu3bojfaddlrgtdclvuxi5gh9aojznm2tbmzkgyqqld+ctuxcji1t9w59+ytrpq1gmxts65l9x7goqzpr3n28qtrk6tb+q3c1vbo7t79v5bq8wpjlroyh7lvoav5uzqumaa01yiky4ctpvb8gbqnx+ovcww93quud/cfcfcrra2utc+8krxqenwgmpldixmq+yvauwuxxqqzgxombgzkuigr2t/dxoxssmqnofyqbbrjnofy6kz4xrs6kskjpgmcz+2dru4osofz/afofoj9fays1nco5n6e2kmi6vguwa6i6whatgbiv957vshl/6yistvvjd5r2hkky7rnazuy5iszuegyckz2rwrazayabnzwytglp68tbpnfdepuhfnxwotiympx3acjxdhaqpwcx7ugcajpmmrvflp1ov1bn3mw3nwnnmif2b9/gdsbzk9</latexit> <latexit 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sha1_base64="/nphngmlolebnzx4h3in2trekpi=">aaab9hicbzdlssnafizp6q3ww9wlm8eitjusikabodanywr2am0ok+mkhtqzxjljocq+hxsxirj1ydz5nk7tllt1h4gp/5zdofn7ewdk2/a3lvtb39jcym8xdnb39g+kh0ctfcas0cyjesg7hlaum0gbmmloo5gkopa4bxvj23m9pafssva86gle3qapbfmzwdpybqn881svogtkonivluyqnqqtgpnbcti1+swv3iakcucfjhwr1xxsslsjlportmefxqxohmkyd2nxomabvw6shj1dz8yzid+u5gmnuvf3riidpaabzzodredquty3/6t1y+1fuqktuaypiitffsyrdte8atrgkhlnpwywkczcisgis0y0yalgqncwv7wkrfoqy1ed+4tsrz7fkyctoiuyohajnbidbjsbwcm8wyu8wrprxxq3phatosuboyy/sj5/ainij/s=</latexit> <latexit sha1_base64="/nphngmlolebnzx4h3in2trekpi=">aaab9hicbzdlssnafizp6q3ww9wlm8eitjusikabodanywr2am0ok+mkhtqzxjljocq+hxsxirj1ydz5nk7tllt1h4gp/5zdofn7ewdk2/a3lvtb39jcym8xdnb39g+kh0ctfcas0cyjesg7hlaum0gbmmloo5gkopa4bxvj23m9pafssva86gle3qapbfmzwdpybqn881svogtkonivluyqnqqtgpnbcti1+swv3iakcucfjhwr1xxsslsjlportmefxqxohmkyd2nxomabvw6shj1dz8yzid+u5gmnuvf3riidpaabzzodredquty3/6t1y+1fuqktuaypiitffsyrdte8atrgkhlnpwywkczcisgis0y0yalgqncwv7wkrfoqy1ed+4tsrz7fkyctoiuyohajnbidbjsbwcm8wyu8wrprxxq3phatosuboyy/sj5/ainij/s=</latexit> <latexit sha1_base64="/nphngmlolebnzx4h3in2trekpi=">aaab9hicbzdlssnafizp6q3ww9wlm8eitjusikabodanywr2am0ok+mkhtqzxjljocq+hxsxirj1ydz5nk7tllt1h4gp/5zdofn7ewdk2/a3lvtb39jcym8xdnb39g+kh0ctfcas0cyjesg7hlaum0gbmmloo5gkopa4bxvj23m9pafssva86gle3qapbfmzwdpybqn881svogtkonivluyqnqqtgpnbcti1+swv3iakcucfjhwr1xxsslsjlportmefxqxohmkyd2nxomabvw6shj1dz8yzid+u5gmnuvf3riidpaabzzodredquty3/6t1y+1fuqktuaypiitffsyrdte8atrgkhlnpwywkczcisgis0y0yalgqncwv7wkrfoqy1ed+4tsrz7fkyctoiuyohajnbidbjsbwcm8wyu8wrprxxq3phatosuboyy/sj5/ainij/s=</latexit> Independence Events A and B are independent if the probability that one occurred is not affected by knowledge that the other occurred. P (A \ B) =P (A)P (B) P (A B) =P (A) P (B A) =P (B) 59

60 <latexit sha1_base64="lxees7hhdaz0mfy9kawoqeetmzo=">aaacdnicbvdlsgmxfm3uv62vuzdugqvqn2vgbn0irrfcvrap6axdjpnpqzpjkgsemvql3pgrblwo4ta1o//gtnufth64chlovetee6amku0431zpzxvtfao8wdna3tnds/cpokpkepm2fkzixoguyzsttqaakv4qcupcrrrh6lrwuw9ekir4vr6nxe/qgnoyyqsnfni1lor1l6qdjnkawpuansbl6kvsrozzmiuu2fwn4uwbl4k7j1uwryuwv7xi4cwhxgogloq7tqr9helnmsotipcpkii8qgpsn5sjhcg/n54zgtwjrdaw0htxckr+nshrotq4cu1ngvrqlxqf+j/xz3r84eeup5kmhm8+ijmgtybfnjcikmdnxoyglknzfeihkghrk2dfhoaunrxmoqcn12m4d2fv5tu8jji4asegdlxwdprgfrrag2dwcj7bk3iznqwx6936mlwwrpnmifgd6/mhviuzvq==</latexit> <latexit sha1_base64="lxees7hhdaz0mfy9kawoqeetmzo=">aaacdnicbvdlsgmxfm3uv62vuzdugqvqn2vgbn0irrfcvrap6axdjpnpqzpjkgsemvql3pgrblwo4ta1o//gtnufth64chlovetee6amku0431zpzxvtfao8wdna3tnds/cpokpkepm2fkzixoguyzsttqaakv4qcupcrrrh6lrwuw9ekir4vr6nxe/qgnoyyqsnfni1lor1l6qdjnkawpuansbl6kvsrozzmiuu2fwn4uwbl4k7j1uwryuwv7xi4cwhxgogloq7tqr9helnmsotipcpkii8qgpsn5sjhcg/n54zgtwjrdaw0htxckr+nshrotq4cu1ngvrqlxqf+j/xz3r84eeup5kmhm8+ijmgtybfnjcikmdnxoyglknzfeihkghrk2dfhoaunrxmoqcn12m4d2fv5tu8jji4asegdlxwdprgfrrag2dwcj7bk3iznqwx6936mlwwrpnmifgd6/mhviuzvq==</latexit> <latexit sha1_base64="lxees7hhdaz0mfy9kawoqeetmzo=">aaacdnicbvdlsgmxfm3uv62vuzdugqvqn2vgbn0irrfcvrap6axdjpnpqzpjkgsemvql3pgrblwo4ta1o//gtnufth64chlovetee6amku0431zpzxvtfao8wdna3tnds/cpokpkepm2fkzixoguyzsttqaakv4qcupcrrrh6lrwuw9ekir4vr6nxe/qgnoyyqsnfni1lor1l6qdjnkawpuansbl6kvsrozzmiuu2fwn4uwbl4k7j1uwryuwv7xi4cwhxgogloq7tqr9helnmsotipcpkii8qgpsn5sjhcg/n54zgtwjrdaw0htxckr+nshrotq4cu1ngvrqlxqf+j/xz3r84eeup5kmhm8+ijmgtybfnjcikmdnxoyglknzfeihkghrk2dfhoaunrxmoqcn12m4d2fv5tu8jji4asegdlxwdprgfrrag2dwcj7bk3iznqwx6936mlwwrpnmifgd6/mhviuzvq==</latexit> <latexit sha1_base64="lxees7hhdaz0mfy9kawoqeetmzo=">aaacdnicbvdlsgmxfm3uv62vuzdugqvqn2vgbn0irrfcvrap6axdjpnpqzpjkgsemvql3pgrblwo4ta1o//gtnufth64chlovetee6amku0431zpzxvtfao8wdna3tnds/cpokpkepm2fkzixoguyzsttqaakv4qcupcrrrh6lrwuw9ekir4vr6nxe/qgnoyyqsnfni1lor1l6qdjnkawpuansbl6kvsrozzmiuu2fwn4uwbl4k7j1uwryuwv7xi4cwhxgogloq7tqr9helnmsotipcpkii8qgpsn5sjhcg/n54zgtwjrdaw0htxckr+nshrotq4cu1ngvrqlxqf+j/xz3r84eeup5kmhm8+ijmgtybfnjcikmdnxoyglknzfeihkghrk2dfhoaunrxmoqcn12m4d2fv5tu8jji4asegdlxwdprgfrrag2dwcj7bk3iznqwx6936mlwwrpnmifgd6/mhviuzvq==</latexit> Independence Events A and B are independent if the probability that one occurred is not affected by knowledge that the other occurred. Pr( \ i E i )= Y i Pr(E i ) 60

61 Clicker Question! 61

62 Clicker Question! Roll two dice. Are the following events independent? Let A = first die is 3. Let B = sum is 6 (a) (b) Yes No (c) Neither :) 62

63 Clicker Question! Roll two dice. Are the following events independent? Let A = first die is 3. Let B = sum is 6 (a) (b) Yes No (c) Neither :) 63

64 Clicker Question! Roll two dice. Are the following events independent? Let A = first die is 3. Let B = sum is 6 (a) Yes (b) No (c) Neither :) P(A) = 1/6 P(B) = 5/36 P(B A) = 1/6!= P(B) A: 3,1 3,2 3,3 3,4 3,5 3,6 B: 1,5 2,4 3,3 4,2 5,1 64

65 Clicker Question! Roll two dice. Are the following events independent? Let A = first die is 3. Let B = sum is 6 (a) Yes (b) No (c) Neither :) P(A) = 1/6 P(B) = 5/36 P(B A) = 1/6!= P(B) A: 3,1 3,2 3,3 3,4 3,5 3,6 B: 1,5 2,4 3,3 4,2 5,1 65

66 Clicker Question! Roll two dice. Are the following events independent? Let A = first die is 3. Let B = sum is 6 (a) Yes (b) No (c) Neither :) P(A) = 1/6 P(B) = 5/36 P(B A) = 1/6!= P(B) A: 3,1 3,2 3,3 3,4 3,5 3,6 B: 1,5 2,4 3,3 4,2 5,1 66

67 Clicker Question! Stan has two kids. One of his kids is a boy. What is the likelihood that the other one is also a boy? (a) 0 (b) 1 (c) 1/2 (d) 1/3 67

68 Clicker Question! Stan has two kids. One of his kids is a boy. What is the likelihood that the other one is also a boy? (a) 0 (b) 1 (c) 1/2 (d) 1/3 68