Probability CSC 172 SPRING 2002 LECTURE 12

Probability Space Set of points, each with an attached probability (non- negative, real number) such that the sum of the probabilities is 1 Simplification The number of points if finite, n Each point has equal probability 1/n Example: A deck of playing cards has 52 “points”

Experiments An experiment is the selection of a point in probability space With the “equally likely” simplification, all points have the same chance of being chosen Example: Pick a card, any card...

Events An event is an set of points in a probability space Prob(E), the probability of an event E is the fraction of the points in E Example: E4 E4 = a 4 is dealt. Prob(E 4 ) = 4/52 = 1/13 Example: Eh Eh = a heart is dealt. Prob(E h ) = 13/52 = 1/4

In general Computing the probability of an event E involves two counts 1. The entire probability space 2. The number of points in E

Conditional Probability Given that the outcome of an experiment is known to be some event E, what is the probability that the outcome is also some other event F? Known as the conditional probability of F given E, or Prob(F|E) == the fraction of the points in E that are also in F

Example: Pick a card, any card Probability space contains 52 points E = “a number card is selected” = 36 points (deuce to 10, 4 suits) F = “the card is less than 7” = 20 points (deuce to 6, 4 suits) Of the 36 points in E, 20 are also in F Prob(F|E) = 20/36 = 5/9

Independent Events F is independent of E if Prob(F|E) = Prob(F) It makes on difference whether E occurs Example: E = “suit is hearts” F = “rank is 5” Prob(F|E) = 1/13 Prob(F) = 4/52 = 1/13

Independence goes both ways n3n3 F E n1n1 n2n2 n4n4

n3n3 F E n1n1 n2n2 n4n4

Example E = “suit is hearts” F = “rank is 5” Prob(F|E) = 1/13 Prob(F) = 4/52 = 1/13 Prob(E) = 13/52 = ¼ Prob(E|F) = 1/4

Complementary Events If E is an event ~E is the event “E does not occur” a.k.a E Prob(~E) = 1 – Prob(E) If F is independent of E, then F is independent of ~E

Expected Value F is some function of points in a probability space Sometimes “payoff” EV(f) = the average over points p of f(p) Example: Would you take the following bet? You bet $8, and draw a card from a 52 card deck The house pays you the value of the card in dollars 2=$2, 3=$3, … 9=$9, 10=$10, J=$10,Q=$10,K=$10,A=$11 (“Blackjack values”)

Example Bet Space = 52 cards f = Blackjack value EV(f) = (4*2+4*3+…+4*9+16*10+4*11)/52 EV(f)= 7.31

A tax on stupidity? NYS Lotto For $1 make 2 Picks of 6 numbers 1-59 How much does the jackpot have to be before it makes sense to play? Space = 59*58*57*56*55*54 = 32,441,381,280 EV(NYSlotto) = payoff/space So, when jackpot is >= $17B, EV(NYSL) >=.50 When the jackpot is $10M, EV(NYSL) about

Well, not exactly In NYS Lotto: First Prize 75.00% Second Prize 7.25% Third Prize 5.50% Fourth Prize 6.25% Fifth Prize 6.00% Also, you may have to pay 50% in taxes don’t forget lawyers, bodyguards, etc. So, now, what does the jackpot have to be? about $45B

Other considerations Entertainment value Charity aspect - “good for education” Social disparity issues Gambling addiction

Counting trick Describe the desired objects by a sequence of steps in which a choice is made from some number of options. Example: The California lottery has announced that every ticket with 3 out of 6 numbers correct will win a “chance to win a” trip to Mexico. What are the odds of selecting exactly 3 numbers (1-53) correct out of 6?

Basic techinique Count the total number of possibilities Count the number of successful possibilites Take the quotient

Basic technique Total =  (53,6) = 53!/47! = 16,529,385,600 Select 3 out of 6 a) Select 3 of the positions for the right guesses b) Select 3 of the 6 numbers drawn c) Select 3 out of 47 wrong numbers a = 6 choose 3= 20, b =P(6,3) = 120, c = P(47,3) = 97, * 120 * = 233,496,000 Prob(“winning”) = 233,496,000/16,529,385,600 = 1.4%