1 / 12

Lecture 14

Lecture 14. 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. A C : every person has a different birthday out of the 21 people.

sancho
Download Presentation

Lecture 14

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lecture 14

  2. 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

  3. Lecture 14

  4. Birthday problem • What about the probability of exactly one pair? • n*(n-1)/2*(365/365)(1/365)(364/365)…(365-n+2)/365

  5. 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?

  6. Play on-line • http://math.ucsd.edu/~crypto/Monty/monty.html

  7. 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

  8. 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)

  9. Calculation • Draw a tree – explain the situation

  10. 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

  11. 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?

  12. 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?

More Related