Math/Stat 341: Probability Second Lecture
Pre-Class ItemsClicker QsMath/Stat 341: ProbabilitySecond LectureSteven J MillerWilliams thematics/sjmiller/public html/341Bronfman 105Williams College, September 14, 2015Hoops Game
Pre-Class ItemsHomeworkClicker QsHoops Game
Pre-Class ItemsHomeworkClicker QsHoops Game
Pre-Class ItemsClicker QsHoops GameThe day. https://www.youtube.com/watch?v uAsV5-Hv-7UBuddy Holly:https://www.youtube.com/watch?v YwHrx0r0t2shttps://www.youtube.com/watch?v GMezwtB1oCUhttps://www.youtube.com/watch?v ku5UeUT7yIQRitchie Valenshttps://www.youtube.com/watch?v Jp6j5HJ-Cokhttps://www.youtube.com/watch?v -ziSLGVQOSghttps://www.youtube.com/watch?v HMcHbh6HBDkThe Big Bopperhttps://www.youtube.com/watch?v 4b-by5e4saIhttps://www.youtube.com/watch?v 3NMklxiE6xw
Pre-Class ItemsClicker QsWho are they and why are they being shown?Hoops Game
Pre-Class ItemsClicker QsJefferson, Adams and Monroe: July 4Hoops Game
7Pre-Class ItemsClicker QsWho are they and why are they being shown?Hoops Game
Pre-Class ItemsClicker QsTruman and Ford, December 26Hoops Game
Pre-Class ItemsClicker QsClicker ProblemsHoops Game
Pre-Class ItemsClicker QsBirthday Problem IBirthday ProblemHow large must N be for there to be at least a 50%probability that two of the N people share a birthday?Hoops Game
Pre-Class ItemsClicker QsBirthday Problem IBirthday ProblemHow large must N be for there to be at least a 50%probability that two of the N people share a birthday?(A) 11 people(B) 22 people(C) 33 people(D) 44 people(E) 90 people(F) 180 people(G) 365 people(H) 500 people.Hoops Game
Pre-Class ItemsClicker QsHoops GameBirthday Problem IBirthday ProblemHow large must N be for there to be at least a 50%probability that two of the N people share a birthday?Probability1.00.80.60.40.21020304050n
Pre-Class ItemsClicker QsHoops GameBirthday Problem IIHow large must N be for there to be at least a 50%probability that two of N Plutonians share a birthday?‘Recall’ one Plutonian year is about 248 Earth years (or90,520 days).(A) 110 people(B) 220 people(C) 330 people(D) 440 people(E) 1,000 people(F) 5,000 people(G) 10,000 people(H) 20,000 people(I) more than 30,000 people.
Pre-Class ItemsClicker QsHoops GameBirthday Problem IIHow large must N be for there to be at least a 50%probability that two of N Plutonians share a birthday?‘Recall’ one Plutonian year is about 248 Earth years (or90,520 days).Probability1.00.80.60.40.2200400600800n
Pre-Class ItemsClicker QsHoops GameProbability QuantitiesPn : Probability at least two share birthday when n in room.Qn : Probability that no shared birthday with n in room.
Pre-Class ItemsClicker QsHoops GameProbability QuantitiesPn : Probability at least two share birthday when n in room.Qn : Probability that no shared birthday with n in room.Note Qn 1 Pn ; oftentimes one is easier to computethan other.What is the correct model?
Pre-Class ItemsClicker QsAssumptions for Birthday ProblemAssume all days equally likely to be birthday, peopleindependent, and the “no February 29th policy”.Hoops Game
Pre-Class ItemsClicker QsHoops GameAssumptions for Birthday ProblemAssume all days equally likely to be birthday, peopleindependent, and the “no February 29th policy”.Gladwell: Canadian Junior Hockey Championship: March11 starts around one side of the Tigers’ net, leaving thepuck for his teammate January 4, who passes it toJanuary 22, who flips it back to March 12, who shootspoint-blank at the Tigers’ goalie, April 27. April 27 blocksthe shot, but it’s rebounded by Vancouver’s March 6. Heshoots! Medicine Hat defensemen February 9 andFebruary 14 dive to block the puck while January 10 lookson helplessly. March 6 scores!.”
Pre-Class ItemsClicker QsExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Hoops Game
Pre-Class ItemsClicker QsExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Q1 365365Hoops Game
Pre-Class ItemsClicker QsExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Q2 365 364·365 365Hoops Game
Pre-Class ItemsClicker QsExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Qn 365 364365 (n 1)····.365 365365Hoops Game
Pre-Class ItemsClicker QsHoops GameExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Qn 365 364365 (n 1)····.365 365365Rewrite:Qn n 1Y k 0k1 365 .
Pre-Class ItemsClicker QsHoops GameExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Qn 365 364365 (n 1)····.365 365365Rewrite:Qn n 1Y k 0k1 365Pavlovian Response: See product, .
Pre-Class ItemsClicker QsHoops GameExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Qn 365 364365 (n 1)····.365 365365Rewrite:Qn n 1Y k 0k1 365 .Pavlovian Response: See product, take logarithm!"n 1 # n 1YXkklog Qn log1 log 1 .365365k 0k 0
Pre-Class ItemsClicker QsHoops GameExpansions: Qn : no shared birthday n in roomQ1 1 (not too surprising!).Qn 365 364365 (n 1)····.365 365365Rewrite:Qn n 1Y k 0k1 365 .More generally if D days in a year:log Qn (D) n 1Xk 0 klog 1 .D
Pre-Class ItemsClicker QsWhy Calculus?Calculus: Linearize the non-linear.Taylor Expansion:log(1 u) u u2 u3 · · · u.23Hoops Game
Pre-Class ItemsClicker QsWhy Calculus?Calculus: Linearize the non-linear.Taylor Expansion:log(1 u) u log Qn (D) n 1Xk 0u2 u3 · · · u.23 klog 1 DHoops Game
Pre-Class ItemsClicker QsHoops GameWhy Calculus?Calculus: Linearize the non-linear.Taylor Expansion:log(1 u) u log Qn (D) n 1Xk 0 n 1Xk 0u2 u3 · · · u.23 klog 1 D n 1k1X kDDk 0
Pre-Class ItemsClicker QsHoops GameWhy Calculus?Calculus: Linearize the non-linear.Taylor Expansion:log(1 u) u log Qn (D) n 1Xk 0 n 1Xk 0 u2 u3 · · · u.23 klog 1 D n 1k1X kDDk 01 (n 1/2)21 (n 1)nor .D2D2
Pre-Class ItemsClicker QsEstimating SolutionLet n1/2;D be value so that probability is 1/2. Then(n1/2;D 1/2)21log Qn1/2;D (D) log ,22DHoops Game
Pre-Class ItemsClicker QsEstimating SolutionLet n1/2;D be value so that probability is 1/2. Then(n1/2;D 1/2)21log Qn1/2;D (D) log ,22Dwhich impliesn1/2;D p2D log 2 p11 D log 4 .22Hoops Game
Pre-Class ItemsClicker QsHoops GameEstimating SolutionLet n1/2;D be value so that probability is 1/2. Then(n1/2;D 1/2)21log Qn1/2;D (D) log ,22Dwhich impliesn1/2;D p2D log 2 p11 D log 4 .22When D 365 predict 22.99, stupendously close to 23!
Pre-Class ItemsMathematica CodeClicker QsHoops Game
Pre-Class ItemsClicker QsVoting: Democratic PrimariesDuring the Democratic primaries in 2008, Clinton andObama received exactly the same number of votes inSyracuse, NY. How probable was this?Hoops Game
Pre-Class ItemsClicker QsHoops GameVoting: Democratic PrimariesDuring the Democratic primaries in 2008, Clinton andObama received exactly the same number of votes inSyracuse, NY. How probable was this? (Note: they eachreceived 6001 votes.)(A) 1 / 10(B) 1 / 100(C) 1 / 1,000(D) 1 / 10,000(E) 1 / 100,000(F) 1 / 1,000,000 (one in a million)(G) 1 / 1,000,000,000 (one in a billion).
Pre-Class ItemsClicker QsHoops GameVoting: Democratic Primaries (continued)Syracuse University mathematics Professor Hyune-JuKim said the result was less than one in a million,according to the Syracuse Post-Standard, which quotedthe professor as saying, “It’s almost impossible.” Hercomments were reprinted widely, as the Associated Presspicked up the story. (Carl Bialik, WSJ, 2/12/08)
Pre-Class ItemsClicker QsHoops GameVoting: Democratic Primaries (continued)Syracuse University mathematics Professor Hyune-JuKim said the result was less than one in a million,according to the Syracuse Post-Standard, which quotedthe professor as saying, “It’s almost impossible.” Hercomments were reprinted widely, as the Associated Presspicked up the story. (Carl Bialik, WSJ, 2/12/08)Far greater than 1/137! What’s going on?
Pre-Class ItemsClicker QsHoops GameVoting: Democratic Primaries (continued)Syracuse University mathematics Professor Hyune-JuKim said the result was less than one in a million,according to the Syracuse Post-Standard, which quotedthe professor as saying, “It’s almost impossible.” Hercomments were reprinted widely, as the Associated Presspicked up the story. (Carl Bialik, WSJ, 2/12/08)Far greater than 1/137! What’s going on?Prof. Kim’s calculation . was based on the assumptionthat Syracuse voters were likely to vote in equalproportions to the state as a whole, which went for Ms.Clinton, its junior senator, 57%-40%. . Prof. Kim saidshe had little time to make the calculation, so she madethe questionable assumption . for simplicity.
Pre-Class ItemsClicker QsFrom Shooting Hoopsto the Geometric Series FormulaHoops Game
Pre-Class ItemsClicker QsSimpler Game: HoopsGame of hoops: first basket wins, alternate shooting.Hoops Game
Pre-Class ItemsClicker QsHoops GameSimpler Game: Hoops: Mathematical FormulationBird and Magic (I’m old!) alternate shooting; first basketwins.Bird always gets basket with probability p.Magic always gets basket with probability q.Let x be the probability Bird wins – what is x?
Pre-Class ItemsClicker QsSolving the Hoop GameClassic solution involves the geometric series.Break into cases:Hoops Game
Pre-Class ItemsClicker QsSolving the Hoop GameClassic solution involves the geometric series.Break into cases:Bird wins on 1st shot: p.Hoops Game
Pre-Class ItemsClicker QsSolving the Hoop GameClassic solution involves the geometric series.Break into cases:Bird wins on 1st shot: p.Bird wins on 2nd shot: (1 p)(1 q) · p.Hoops Game
Pre-Class ItemsClicker QsHoops GameSolving the Hoop GameClassic solution involves the geometric series.Break into cases:Bird wins on 1st shot: p.Bird wins on 2nd shot: (1 p)(1 q) · p.Bird wins on 3rd shot: (1 p)(1 q) · (1 p)(1 q) · p.
Pre-Class ItemsClicker QsHoops GameSolving the Hoop GameClassic solution involves the geometric series.Break into cases:Bird wins on 1st shot: p.Bird wins on 2nd shot: (1 p)(1 q) · p.Bird wins on 3rd shot: (1 p)(1 q) · (1 p)(1 q) · p.Bird wins on nth shot:(1 p)(1 q) · (1 p)(1 q) · · · (1 p)(1 q) · p.
Pre-Class ItemsClicker QsHoops GameSolving the Hoop GameClassic solution involves the geometric series.Break into cases:Bird wins on 1st shot: p.Bird wins on 2nd shot: (1 p)(1 q) · p.Bird wins on 3rd shot: (1 p)(1 q) · (1 p)(1 q) · p.Bird wins on nth shot:(1 p)(1 q) · (1 p)(1 q) · · · (1 p)(1 q) · p.Let r (1 p)(1 q). Thenx Prob(Bird wins)p rp r 2 p r 3 p · · · p 1 r r2 r3 · · · ,the geometric series.
Pre-Class ItemsClicker QsSolving the Hoop Game: The Power of PerspectiveShowedx Prob(Bird wins) p(1 r r 2 r 3 · · · );will solve without the geometric series formula.Hoops Game
Pre-Class ItemsClicker QsSolving the Hoop Game: The Power of PerspectiveShowedx Prob(Bird wins) p(1 r r 2 r 3 · · · );will solve without the geometric series formula.Havex Prob(Bird wins) p Hoops Game
Pre-Class ItemsClicker QsSolving the Hoop Game: The Power of PerspectiveShowedx Prob(Bird wins) p(1 r r 2 r 3 · · · );will solve without the geometric series formula.Havex Prob(Bird wins) p (1 p)(1 q)Hoops Game
Pre-Class ItemsClicker QsSolving the Hoop Game: The Power of PerspectiveShowedx Prob(Bird wins) p(1 r r 2 r 3 · · · );will solve without the geometric series formula.Havex Prob(Bird wins) p (1 p)(1 q)xHoops Game
Pre-Class ItemsClicker QsHoops GameSolving the Hoop Game: The Power of PerspectiveShowedx Prob(Bird wins) p(1 r r 2 r 3 · · · );will solve without the geometric series formula.Havex Prob(Bird wins) p (1 p)(1 q)x p r x.
Pre-Class ItemsClicker QsHoops GameSolving the Hoop Game: The Power of PerspectiveShowedx Prob(Bird wins) p(1 r r 2 r 3 · · · );will solve without the geometric series formula.Havex Prob(Bird wins) p (1 p)(1 q)x p r x.Thus(1 r )x p or x p.1 r
55Pre-Class ItemsClicker QsHoops GameSolving the Hoop Game: The Power of PerspectiveShowedx Prob(Bird wins) p(1 r r 2 r 3 · · · );will solve without the geometric series formula.Havex Prob(Bird wins) p (1 p)(1 q)x p r x.Thus(1 r )x p or x p.1 rAs x p(1 r r 2 r 3 · · · ), find1 r r2 r3 · · · 1.1 r
Pre-Class ItemsClicker QsLessons from Hoop Problem Power of Perspective: Memoryless process. Can circumvent algebra with deeper understanding!(Hard) Depth of a problem not always what expect. Importance of knowing more than the minimum:connections. Math is fun!Hoops Game
Williams College, September 14, 2015 1. Pre-Class Items Clicker Qs Hoops Game Homework 2. Pre-Class Items Clicker Qs Hoops Game Homework 3. . (C) 33 people (D) 44 people (E) 90 people (F) 180 people (G) 365 people . Pre-Class Items Clicker Qs Hoops Game Solving the Hoop Game
Joint Probability P(A\B) or P(A;B) { Probability of Aand B. Marginal (Unconditional) Probability P( A) { Probability of . Conditional Probability P (Aj B) A;B) P ) { Probability of A, given that Boccurred. Conditional Probability is Probability P(AjB) is a probability function for any xed B. Any
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Math 5/4, Math 6/5, Math 7/6, Math 8/7, and Algebra 1/2 Math 5/4, Math 6/5, Math 7/6, Math 8/7, and Algebra ½ form a series of courses to move students from primary grades to algebra. Each course contains a series of daily lessons covering all areas of general math. Each lesson
