Understanding the Birthday Problem and Monty Hall Challenge
This lecture explores the concepts of probability using two engaging problems: the Birthday Problem and the Monty Hall Problem. We analyze the likelihood of at least two people sharing a birthday in a class of 21 and derive the probability through complementary counting. We also delve into the Monty Hall Problem, evaluating the strategic choices available after a contestant selects a door and the host reveals a non-winning door. Both problems illuminate foundational principles of probability and decision-making in uncertain scenarios.
Understanding the Birthday Problem and Monty Hall Challenge
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Presentation Transcript
Birthday Problem • In a classroom of 21 people, what is the probability that at least two people have the same birthday? • Event A: at least two people have the same birthday out of the 21 people. • AC: every person has a different birthday out of the 21 people. • P(A)=1-P(AC) =1-(365/365)(364/365)…(345/365) Lecture 14
Birthday problem • What about the probability of exactly one pair? • n*(n-1)/2*(365/365)(1/365)(364/365)…(365-n+2)/365
Monte Hall Problem • 3 doors, one prize • Select one door • Host show opens one of the other two doors that do not contain the prize • You are given a chance to keep the door you selected or switch to the other non-open door. • What shall I do?
Play on-line • http://math.ucsd.edu/~crypto/Monty/monty.html
Analysis • Assumptions: • Initially, each door has the same chance to contain the price • If selected door contains the price, Monty selects the door to open at random with equal probability
Setup is important • I can relabel the doors: • M– the one I selected • L– left door out of the remaining • R– right door out of the remaining • P(Prize in M)=P(prize in L)=P(prize inR)=1/3 • Two events: Open L, Open R • We need P(Prize in M | Open L)
Calculation • Draw a tree – explain the situation
Modifications • Possible modification: • Monty favors a door: What changes is P(Open L | Price in R) ≠ 1/2 • Monty can goof (open a door with the price in it)The tree changes • In any case switching never hurts
Limitation of mean • When evaluating games – we often looked at the mean gain as a proxy for understanding the game • This might be insufficient • In magamillions and powerball the jackpot sometimes rises so high that the average gain is positive. Q: Is it rational to play? • Issues: • Adjustment for ties (drops down expected gain significantly) • How many games one needs to play before winning?
Let’s design a Lottery! • How to make a lottery? • Define random generating mechanism • Define payoffs • Makes money on average • Risk is not too bad • How much reserves are needed?
