Midterm Review, Math 210G-04, Spring 2011

1. Midterm Review, Math 210G-04, Spring 2011

2. The barbers paradox asks the following: A) In a village, the barber shaves everyone who does not shave himself, but no one else. Who shaves the barber? B) In a town, one barber cannot stay in business but two barbers can C) If every man except the barber shaves himself, who shaves the barber? D) If every man except the barber shaves himself, how can the barber stay in business?

3. Zeno’s paradox of achilles and the tortoise asserts that A) In a race between Achilles and the tortoise, Achilles can never pass the tortoise B) In a race between Achilles and the tortoise, Achilles will never win C) In a race between Achilles and the tortoise, the race ends as soon as Achilles and the tortoise are tied D) Achilles is afraid to race the tortoise

4. Russell’s paradox implies that A) There cannot be a set that contains all other sets B) A set that contains only apples cannot contain any oranges C) A set cannot contain itself D) This sentence is false

5. A tautology is a statement that is A) Always true B) Always tight C) redundant

6. The Pythagoreans were divided int “mathematikoi” and “akousmatikoi” • A) True • B) False

7. Dice are descendents of • A) pottery • B) rubiks cube • C) dodecahedrons • D) buckminsterfullereen • E) bones

8. A fair coin is defined as one that has a probability of ½ of coming up heads • A) True • B) False

9. The statement ((pq)∧p)q is A) Always true B) True only when p is true C) True only if q is true D) True only if both p and q are true

10. In the Monte Hall problem, if a car is behind one of three doors and, after you choose one door Monte shows a goat behind one of the others, then you should always switch to the other door. • A) True • B) False

11. What is the intended conclusion of the following • All writers, who understand human nature, are clever. (W -> C) • No one is a true poet unless he can stir the hearts of men. (~S -> ~ P) • Shakespeare wrote “Hamlet”. (WH <-> SH) • No writer, who does not understand human nature, can stir the hearts of men. (~W->~S) • None but a true poet could have written “Hamlet”. (~P->~WH)

12. Determine the intended implication of the following collection of statements: • (a) No interesting poems are unpopular among people of real taste. • (b) No modern poetry is free from affectation. • (c) All your poems are on the subject of soap-bubbles. • (d) No affected poetry is popular among people of real taste. • (e) No ancient poem is on the subject of soap-bubbles.

13. Explain why at least one of the following pictures proves the Pythagorean theorem

14. Propose a solution to the following problem: A king decides to give 100 of his prisoners a test. If they pass, they can go free. Otherwise, the king will execute all of them. The test goes as follows: the prisoners stand in a line, all facing forward. The king puts either a black or a white hat on each prisoner. The prisoners can only see the colors of the hats in front of them. Then, in any order they want, each one guesses the color of the hat on their head. Other than that, the prisoners can not speak. To pass, no more than 1 of them may guess incorrectly. If they can make their strategy before hand, how can they be assured that they will survive?

15. Compute the probabilities of the following outcomes for rolling a pair of dice • The sum of the dice is odd • The sum of the dice is larger than 6 • The sum of the dice is less than 1 • The sum of the dice is less than 6

16. Bayes rule states that Find P(A|B) if P(B|A)=1/2, P(A)=1/3 and P(B)=2/3.

17. Bayes rule, part II • Professor Doolittle conducted a survey and discovered that 80% of the students who got an A on his exam studied for it for 4 or more hours. • He also learned that half of his students studied for 4 or more hours for the exam. • 20 % of Doctor Doolittle’s students get an A on the exam. • Use Bayes’ rule to determine the probability that someone who studies for 4 or more hours the day before will get an A on his exam.

18. Fill in the next row of Pascal’s triangle

19. How many ways are there to choose a subset of 4 elements from a set of six elements? • A) 30 • B) 15 • C) 4 • D) 6 • E) None of the above

20. Poker hands • Compute the number of possible ways of choosing 5 cards from a deck of 52 cards. • Compute the number of possible ways of getting 4 of a kind in a five card poker hand. Explain your result and its probability of happening.

21. 3 card guts • Three card cuts is a version of poker in which each player gets three cards. Straights and flushes are not allowed. The best possible hand is three aces. • What is the number of distinct hands in 3 card guts? • How many of these hands allow three of a kind? • How many allow a pair (but not three of a kind)? • How many allow no pairs?

22. Find the average of the following numbers and their standard deviation • The variance of numbers x1 ,…, xNis the sum of the squares of their differences from their mean, divided by N-1. • The sample deviation is the square root of the variance. • The numbers are: 72, 66, 70, 54, 60, 78, 72, 64, 66, 56, 82