Paradox

1. One of 3 prisoners is to be executed; the other 2 to go free. Prisoner A ask the jailer for the one name from the other two to be set free claiming no harm because he knows at least one will go free The jailer refuses claiming A’s probability of being executed would rise from 1/3 to ½. Who’s right?

P(A) = P(B) = P(C) = 1/3; Say Jailer claims B will be freed.

~B => B will not be executed where P(~B) = 1

Then P(A | ~B) = P(A, ~B)/ P(~B)

=

P(A|~B) = P(A)*P(~B|A) = (1/3)* 1. P(C) = 2/3.

(1/ 3)*1) 11 3

05/06/23 1rd

Behind 3 door are 2 goats and the car of your dreams. The host knows what is behind each door. You pick a door but the door remains closed. The host then opens one of the remaining 2 doors and shows you a goat. He allows you to keep your original door or to switch to the remaining door. What do you do?

A B C

05/06/23 2rd

A and B are baseball players. B bats twice A’s average during the first half of the season and again twice A’s average during the second half of the season. At the end of the season, is it possible for A’s average to be higher than B’s?

1st Half 2nd Half Season

A 100/400 = .250 2/20 = .100 102/420

B 10/20 = .500 80/400 = .200 90/420

05/06/23 3rd

St Petersburg Paradox

You are awarded $2x if the event heads does not occur until the xth flip. Compute expected value.

E(X) = ½ *21 + ¼ * 22 + … = 1 + 1 + 1 + … + = .

X 2 4 8 16 32 64 …P(X) ½ ¼ 1/8 1/16 1/32 1/64 …

05/06/23 4rd

Urn with 1 W and ? R

An urn has 1 white and 1 red ball. A ball is randomly selected. If W, game over. If R, the R is put back and an R is added. Repeat trial. Find E(X) where X is number of picks at which W occurs.1W 1 R ½ = P(W)1W 2 R 1/3 = P(W) = ½ * 2/31W 3 R ¼ = P(W)E(X) = ½ + 1/3 + ¼ + … + 1/n + … = Suppose you wanted to see 1000 R in urn.Try (sim-urn-wr 1000 2000)

05/06/23 5rd

Wallet Paradox

You randomly pick a person's wallet to see how much money it contains. If more than yours, you keep it. If less than yours, you must give him your wallet's content. You are opposing players; yet each stands to gain more than each risks. This zero sum game appears favorable to both players, which is impossible.

(sim-wallet 100 10) $100 max and 10 bets((88 97 64 74) (91 74 51 4 47 50)) summing to (323, 317) respectively.

05/06/23 rd 6

2 envelope paradox

A player is given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. The player may select one envelope and keep whatever amount it contains, but upon selection is offered the possibility to take the other envelope instead.Either $A & 2A or A & A/2Thus, the other envelope contains 2A with probability 1/2 and A/2 with probability1/2Expected value ½ * A/2 + ½ * 2A = 5A/4 in other envelope suggesting switching. Now reason after switching and it again appears to switch back.

05/06/23 rd 7

Elevator Paradox

In a 20-floor building with one elevator, with you on the 5th floor, what is the likely direction of the elevator when it arrives on your floor?

Continuous uniform implies more likely elevator is above you coming down than below you going up.

05/06/23 rd 8

Liar Paradox

Cellini always makes a sign with a false statement; Bellini with a true statement.You come across a sign that reads

THIS SIGN WAS MADE BY CELLINI.Who made the sign?

05/06/23 rd 9

Train Paradox

East and West bound trains run at 10-minute intervals A stud goes to the tracks daily to catch the first train

arriving at the station to see his girlfriends. One is happy to see him 9 out 10 days and the other is angry seeing him just 1 out of 10 days. Why so?

Schedule of East bound: 12:00 12:10 12:20 …Schedule of West bound: 12:01 12:11 12:21 …

05/06/23 rd 10