E-320: Teaching Math with a Historical Perspective Oliver Knill, 2011Lecture 8: Probability1. CombinatoricsHere are themost impo r tantcombinatoricsproblems:How many ways are there to: The answer is:permute n elements n! = n(n − 1)...2 · 1choose k fro m n with repetitions nkpick k different from n if order mat t ersn!(n−k)!pick k different from n without looking at order nk!=n!k!(n−k)!Problem 1: You are playing ”scrabble”. You are stuckwith the letters O, L, I, V, E, R. How many single wordsof length 6 can you write?Problem 2: how many novels can one write of length338’787, the length of the ”Hound of Baskervilles”. Youcan assume an alphabet of 30 including space and punc-tuations.Problem 3: How many ways are there to chose 3 peo-ple fro m a contestant group of 12 if the order does notmatter?Problem 4: A combination lock has 40 numbers 0−39.A lock combination consists of 3 different numbers,where the order matters. How many different lock com-binations are there?The Monty Hall ProblemWe want to understand the famous Monty Hall problemYou have to choose from three doors. Behind one door isa car and behind the others are goats. You pick a door.The host, who knows what’s behind the doors, opensan another doo r, one which has a goat. You have thechoice to choose the door or to switch. What is better?The ”three door problem” is also called ”Monty Hall problem” and became a sensation andcontroversy in 1991. Intuitive argumentation can lead to the conclusion that it do es not matterwhether to change the door or not. When asked, a larg e major ity of test persons tell that it doesnot matter.1) We first assume that we decide not to switch.You choose a door. Note that the revelation of the host does not affect your choice.What is the probability that you win in this case?2) Now we switch. We look at three possibilities now.What happens if you initially chose the door with the car? Do you win or lose in this case?3) What happens if you init ially chose the door with the goat? Do you win or lose in this case?We have computed the winning probability with no switching and then in 2-3) the winning prob-ability with switching. What do you conclude?4) For the following question, most people would say 1/2.Dave has 2 kids. One of them is a boy. What is theprobability that the other kid is a girl?The Bertrand paradoxThe Bertrand paradox illustrates that one has to be clear on how to setup a probabilisticmodel in a concrete situation.We t hrow ra ndo m lines onto the unit disc. Whatis the probability that the line intersects the discwith a length ≥√3, the length of the inscribedequilateral triangle?Problem 1) Take an arbitrary point on theboundary of the disc. The set of all lines throughthat point are parameterized by an angle φ. Forwhich midpoints is the length of the chord longerthan the equilateral triangle length? By compar-ing angles, what is the probability?Problem 2) Take now all lines perpendicular toa fixed diameter. The entire diameter has length2. Where do the chords hit the diameter so thatit is longer than√3? By comparing lengths, whatis the probability?Problem 3) Look at the midpoints of the chords.Where does such a midpoint have to be so thatthe chord is longer than√3? By comparing areas,what is the probability now?The Petersburg ParadoxThe origins of probability was in gambling. We look here closely at the Petersburg paradox, whichhad been devised by Daniel Bernoulli in 1738. You pay a fixed entrance fee C and you get theprize 2T, where T is the number of times, the casino flips a coin until ”head” appears.For example, you enter 10 dollars. If the sequence of coin experiments would give ”tail, tail, tail,head”, you win 23− 10 = 8 − 10 = −2 dollars. This means you have lost 2 dollars in this game.1) Build groups of 2-4 . One is the casino, the others play the casino. Choose an entrance feewhich you think is fair and play as many times as time allows. In the end, record your winning.2) What is the probability that you lose your entire winning? That is, what is t he chance that wehave ”head” the first time? Note that T = 0 in this case.3) What is the probability that we have ”head” the second time? Note t hat T = 2 in this case.How much do we win or lose in this case?4) What is the probability t ha t ”head” appears the third time? Note tha t T = 3 in this case.How much did you win or lose in this case?5) What is the probability tha t ”head” appears at time T = n the first time? How much did youwin or lose in this case?Fair would be an entrance fee which is equal to the expectation of the win, which is 1 · P[T =0] + 2 · P [T = 1] + 5 · P[T = 2] + ...... What does ”fair” mean? For example, the situationT = 20 is so improbable that it never occurs in the life-time of a person. Therefore, for anypractical reason, one has not to worry about large values of T . This, as well as the finitenessof money resources is the reason, why casinos do not have to worry about the following bul-let proof martingale strategy in roulette: bet c dollars on red. If you win, stop, if you lose,bet 2c dollar s on red. If you win, stop. If you lose, bet 4c do llars on red. Keep doubling thebet. Eventually after n steps, red will occur and you will win 2nc−(c+2c+···+2n−1c) = c dollars.How does one solve the Petersburg paradox? What would be a reasonable entrance fee in ”reallife”? Bernoulli proposed to replace the expectation E[G] of the profit G = 2Twith the expectation(E[√G])2, where u(x) =√x is called a utility function. This would lead to a fair entrance(E[√G])2= (∞Xk=12k/22−k)2=1(√2 − 1)2∼ 5.828... .Similar effects appear in polit ical situations as in voting systems, where different voting systemscan produce different winners. The following example is by Donald Saari:”Consider 15 people deciding what beverage to serve at a party. Six prefer milk first, wine second,and beer third; five prefer beer first, wine second, and milk third; and four prefer wine first, beersecond, and milk third. In a plurality vote, milk is the clear winner. But if the gro up decidesinstead to hold a runoff election between the two top contenders milk and beer, then beer wins,since nine people prefer it over milk. And if the group awards two points to a drink each time avoter ranks it first and one point each time a voter ranks it second, suddenly wine is the