Midterm Review, Math 210G-07, Spring 2009

1) Explain why the following pictures prove the Pythagorean theorem

2) 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)

3) The following Escher picture is an example of

• Hyperbolic geometry• Euclidean geometry

4) 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?

5) 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

6) Bayes rule states that

• Find P(A|B) is P(B|A)=1/2, P(A)=1/3 and P(B)=2/3.

7) 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 4 or more hours the day before.

• He also learned that half of his students studied for 4 or more hours the day before.

• 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.

8) Fill in the next row of Pascal’s triangle

1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

9) 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.

10) Find the average of the following numbers and their standard deviation

• The variance of numbers x1 ,…, xN is 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 number are: 72, 66, 70, 54, 60, 78, 72, 64, 66, 56, 82

11) Use the Euclidean algorithm to find the following:

• GCD(69,96)• GCD(35,53)• GCD(123,321)

12) Which of the following numbers are prime numbers?

• 11,101,1001,10001• 13,31,113,311,1331• 41,141,1441• Here are some divisibility tests