Assignment 79

"And since geometry is the right foundation of all painting, I have decided to teach its rudiments and principles to all youngsters eager for art." - (Albrecht Durer, 1471-1528)

Math History Tidbit:

Ethnomathematics: This term relates to the study of the mathematical experiences of those from different countries, from different ethnic and social backgrounds, or of a gender different from yours. Mathematics is a universal subject, but the ways different groups learn it, or experience it, very considerably. In 1985, the International Study Group on Ethnomathematics (ISGEm) was created. This group actively promotes programs to improve conditions for mathematics study by groups not usually encouraged to work with mathematics or mathematical ideas. With the continued work of ISGEm, mathematics educators around the world are finding that informal mathematics education is pervasive, that the social context of education has a great influence on student performance, and that quality education depends upon an understanding of culture.

Herkimer's Corner

What did Herkimer say when he was informed that he was wearing one red sock and one green sock?

Answer: "I have another pair at home just like this pair."

Herky wants to know:

If you offer your child $10 for passing a mathematics test, can you consider you have saved $10 if he fails the test?

If workers go on strike at the U.S. Mint, it is because they want to make fewer dollars?


Reading: Section 12.5, pages 730-733.

Written: Page 734-735/18-23, 25-29.

Items for reflection:

Mathematical numbers:
PALINDROMIC NUMBER: A positive integer of two or more digits whose value is the same read forwards or backwards. Some examples of palindromic numbers include 33, 777, 4994, and 99399. The year 2002 is a palindromic year.

Two events are independent if the outcome of one of them has no affect on the outcome of the other.


QUESTION: A roll a red die, and then I roll a green die. What is the probability that I get a 6 on the red die and a 6 on the green die?

RESPONSE: The outcome on the green die is not affected by the outcome on the red die. The two rolls are independent. The requested probability is (1/6)(1/6) = 1/36.

QUESTION: I pick a card from a thoroughly shuffled deck of 52 cards, and note what it is. Then, I pick another card without replacing the first card. What is the probability that the second card is a heart?

RESPONSE: The outcome of the second event (picking a second card) is dependent on the first event (picking the first card), since it depends on whether or not the first card was a heart. If the first card was a heart, then the probability that the second card is a heart is 12/51. If the first card was not a heart, then the probability that the second card is a heart is 13/51.

QUESTION: If I roll a single die four times, what is the probability that I get at least one 6?

RESPONSE: Each roll is independent of any other roll. The easiest way to respond to the question asked is to calculate the probability that I get no 6's, then subtract that from 1, or 100%. The answer is 1 - (5/6)4 = 0.5177, or about 51.77%.

QUESTION: The probability that I win a specific game is 5%. How many games would I have to play to have a 75% chance of winning at least once?

RESPONSE: The probability of winning x games in a row is (.95)x. Hence, the probability of winning at least one game when x games are played is 1 - (.95)x. So, we want to solve the equation 1 - (.95)x = .75 This yields (.95)x = .25 ==> x = log.95(.25) = log(.25)/log(.95) = 27.0268. In other words, you would expect to win at least once in 27 games if your probability of winning is just 5%.

Problem: I flip a coin, roll a single die, and randomly choose a digit from the set {0,1,2,3,4,5,6,7,8,9}. What is the probability that the coin comes up heads, the die shows a 6, and the random digit is greater than 5?

Solution (with communication): The three events described are independent. Hence the requested probability is

(1/2)(1/6)(4/10) = 0.0333333, or about 3.3%.

Problem: If I roll a single die ten times, what is the probability that I will get at least one 5?

Solution (with communication): Each roll is independent of any other roll. The probability I would not get a 5 in ten rolls is (5/6)10. Hence the probability that I would get at least one 5 is

1 - (5/6)10 = 0.83849, or approximately 84%.