  Assignment 83

 "I tell them that if they will occupy themselves with the study of mathematics they will find in it the best remedy against the lusts of the flesh." - (Thomas Mann, 1875-1955, The Magic Mountain) George Boole (1815-1864): While a professor of mathematics in Ireland, Englishman Boole began to concentrate on the logic used in the construction of algebraic systems. In his book, Investigation of the Laws of Thought (1854), Boole established the foundations for modern symbolic logic, and in the process he created a new algebra, which we now call Boolean algebra. One of the true mathematical innovators of his century, Boole's ideas are applied in this modern day in circuit design, probability, insurance, and information theory. Why did Herkimer ask a pretty girl to go with him to a barbecue? Answer: He wanted her to be his grill friend. Herky's friends: LIZ ONYA: She really loves Italian food. ARTHUR ITIS: This guy always complains about soreness in his knees. ASSIGNMENT #83 Reading: Section 13.1, pages 769-771. Written: None, but be sure you understand Examples 1-5 presented in the reading. This section is basically a review of the elementary trigonometry you studied in geometry. NOTE: This assignment is due at the class meeting that follows the test on Chapter 12. Mathematical note of interest:Ancient Egyptian arithmetic operations didn't include fractions that contained numerators other than 1. The Rhind Papyrus (ca 1850 B.C.) contains a table that allowed the reader to respresent fractions as a sum of unit fractions (fractions with a numerator of 1). For instance, 2/97 = 1/56 + 1/679 + 1/776.
1. Review: Binomial Setting.

In a given event, I have a 62% chance of success and a 38% chance of failure. I will try the event ten times. If x is the number of times I am successful, calculate the probabilities associated with each value of x.

 Value of x 0 1 2 3 4 5 6 7 8 9 10 Probability (in symbols) . . . . . . . . . . . Probability (%) . . . . . . . . . . .

a) What is the most likely value of x?

b) What is the least likely value of x?

c) What is the probability that I will be successful fewer than 4 times?

d) What is the probability I will be successful less then 8 times and more than 4 times?

e) How many attempts must I make in order to be 50% sure that I will have at least one success?

f) How many attempts must I make in order to be 90% sure that I will have at least one success?

g) Produce a histogram displaying the probabilities associated with each value of x.

h) What is the probability that I will be successful in three consecutive trials?

i) What is the probability that I will fail in three consecutive trials?

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2. In a special form of LOTTO, you pay \$1 to pick 2 numbers from the set S ={1,2,3,...,48,49,50}. At a future date, two numbers are randomly selected from S. If your two numbers match the selected numbers, you win \$10,000.

a) What is the probability you would win if you purchased one ticket?

b) If you purchased one ticket a day, how many days would be required for you to have a 50% chance of matching the two numbers at least once?

 Problem: In a right triangle, one of the acute angles has a measure of q degrees. If sin q = 2/5 and the leg of the right triangle opposite q has length 3.7 inches, what is the length of the hypotenuse of the triangle. Solution (with communication): Let the length of the hypotenuse be x inches. We then have sin q = 2/5 = 3.7/x ==> x = (3.7)(5/2) = 9.25 (in.) Problem: In a right triangle, if A is one of the acute angles, show that (sin A)/(cos A) = tan A. Solution (with communication): Let a, b, and c be the sides of the right triangle with c being the hypotenuse and a being the side opposite angle A. Then sin A = a/c and cos A = b/c. Hence (sin A)/(sin B) = (a/c)/(b/c) = (a/c)(c/b) = a/b = tan A.