CSProbabilitypartbv.pptVIP

* * * * * * * * * * * * * * * * * * * * * * * * * * * * Probability Paradox I Magic Dice: Or How to Win Every Time! a) You select any one of the four dice (A, B, C, or D). b) I’ll select another. Both dice are thrown, highest number wins throw. Do series of 10 throws. The person with the most highest throws wins the series. (I.e. die “more likely to get a higher number” wins.) Claim: In a game of The Best of Ten Throws’, I will almost certainly win --- no matter which die you pick!! Why is this strange? Say, you pick die A. Let’s assume, die B is better. So, I pick B. But, then, next game next person picks B. Let’s assume C is better. I’ll select C. Next person, will pick C. I’ll pick D. Next person, will pick D… Hmm… ?? I’ll pick A and will win!! A B C D …. A !! Failure of transitivity! But, could such a set of dice exist? Surprisingly, yes! A B C D Magic Dice mathtrek.asp D C B A Prob(D wins over C) = 2/3 2/6 + (4/6)* (1/2) = 4/6 Prob(C wins over B) = 2/3 since 3/6 + (3/6)* (2/6) = 4/6 Prob(B wins over A) = 4/6 = 2/3 (i.e. Prob(A wins over B) = 1/3) Prob(A wins over D) = 4/6 = 2/3 A B C D …. A !! How about expected value of dice throw? Transitive or not?? E.g. E[B] = 16/6. YES!! E[B] E[A] = E[C] E[D] 16/6 18/6 = 18/6 20/6 * Random Variables and Distributions Random Variables For a given sample space S, a random variable (r.v.) is any real valued function on S, i.e., a random variable is a function that assigns a real number to each possible outcome Suppose our experiment is a roll of 2 dice. S is set of pairs. Example random variables: S 0 2 -2 X = sum of two dice. X((2,3)) = 5 Y = difference between two dice. Y((2,3)) = 1 Z = max of two dice. Z((2,3)) = 3 Sample space Numbers Random variable Suppose a coin is flipped three times. Let X(t) be the random variable that equals the number of heads that appear when t is the outcome. X(HHH) = 3 X(HHT) = X(HTH) = X(THH) = 2 X(TTH) =