"Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak." - (Isaac Asimov)
William Rowan Hamilton (1805-1865): Hamilton shocked the world of mathematics by inventing a new type of number called a quaternion. A quaternion is a vector-like quantity. The laws of quaternion algebra are the same as those of ordinary algebra - except that the commutative law for multiplication that we take for granted does not hold true. That is, if x and y are quaternions, then it doesn't follow that xy = yx. In fact, in the system of quaternions, xy = (-y)x. Hamilton's creation made mathematicians realize that the laws of common algebra are not universal truths. His work led to the invention of other useful algebras and paved the way for modern abstract algebra.
Where did Herkimer spend most of his time when he was a diamond cutter?
Answer: Mowing grass at the local baseball park.
MEL PRACTICE: This guy was a medical doctor, but not a very good one.
GRACE FULL: A very polished ballet dancer.
Reading: Review text sections on probability, as necessary.
Written: Handout containing problems relating to the binomial setting.
Here is a binomial setting situation we will examine in class:
LARGEST NUMBER THAN CAN BE REPRESENTED USING JUST THE DIGITS 1,2,3,4, THE SYMBOL FOR MINUS (-), AND THE DECIMAL POINT (.). BASICALLY, YOU CAN USE ONLY THE SYMBOLS IN THE SET (-,.,1,2,3,4}. It is believed that the largest number than can be represented using just these six symbols is
This number has 433 digits.
In a manufacturing process, it is permissible that 3% of the items exceed a weight of 32 ounces. Random samples of size 25 are examined. Let x = the number of items in the sample that exceed 32 ounces in weight?
1. What are the possible values of the random variable x?
2. What is the probability that none of the items in a sample of 25 exceed 32 ounces?
3. What is the probability that exactly two items in a sample of 25 exceed 32 ounces?
4. What is the most likely value of the random variable x?
5. Calculate the probabilities for each possible value of the random variable x.
6. Sketch a histogram displaying the expected distribution of the random variable x.
7. Suppose you just started picking items at random from the production line. How many would you expect to have to pick to have
a) a 50% chance of getting an item exceeding 32 ounces in weight?
b) a 95% chance of getting an item exceeding 32 ounces in weight?
Problem: When attempting a particular event, I have a 25% chance of success. I attempt the event three times. Let x represent the number of times I am successful.
(a) What are the possible values for the random variable x?
(b) What are the probabilities for each value of x?
Solution (with communication):
(a) The possible values for x are 0, 1, 2, 3.
(b) Prob(x =0) = 3C0(.25)0(.75)3 = 0.421875, or about 42.19%.
Prob(x =1) = 3C1(.25)1(.75)2 = 0.421875, or about 42.19%.
Prob(x =2) = 3C2(.25)2(.75)1 = 0.140625, or about 14.06%.
Prob(x =3) = 3C3(.25)3(.75)0 = 0.015625, or about 1.56%.
Note: The probabilities add up to 100%, as should be expected.