Assignment 81
"The chief aim of all investigations of the external world should be to discover the rational order and harmony which has been imposed on it by God and which He revealed to us in the language of mathematics."  (Johannes Kepler, 15711630)
In 1897, the Indiana State State Legislature attempted to legislate the value of pi with House Bill Number 246. The wording starts this way:
Be it enacted by the General Assembly of the State of Indiana: It has been found that the circular area is to the quadrant of the circumferences, as the area of an equilateral rectangle is the the square of one side.
The Bill passed the House 670, but, after ridicule, it was shelved by the Senate. In his book, A History of Pi, Petr Beckman writes:
The Bill contains more hairraising statements which not only contradict elementary geometry, but also contract each other.
Among other things, the Bill assumes that if a circle and a square have equal perimeters, then they have equal areas. 
Why did Herkimer want a square bathtub in his house?
Answer: So there wouldn't be a ring of dirt after he took a bath.
Herky's friends:
ELLA PHANT: A young lady who takes care of the large animals at the local zoo.
MISS DEMEANOR: She is a lawyer who handles nonfelony cases. 
ASSIGNMENT #81
Reading: Review Section 12.6, as necessary.
Written: The problems (below) from Assignment #80. These represent binomial settings.
(1) I roll a single die five times, Find the probabilities and distribution associated with the random variable x, where x is the number of 6's obtained. Note that x could be 0,1,2,3,4,5. (2) I flip a coin five times, Find the probabilities and distribution associated with the random variable x, where x is the number of HEADS obtained. Note that x could be 0,1,2,3,4,5. 
Mathematical numbers: LARGEST NUMBER THAN CAN BE REPRESENTED WITH JUST THREE DIGITS: If x = 9^{9}, then the largest number than can be represented with just three digits is 9^{x}. Think about this for a bit: This number is 9 raised to the power 9^{9}, and not 9^{9} raised to the power 9, which would be 9^{81}. 
Data collection, empirical probabilities, and theoretical probabilities associated with the random variable x, where x is the number of 6's obtained when a single die is rolled five times. (Or, when five dice are rolled at once.)
Value of x 
Empirical computations relating to probability 
Theoretical computations relating to probability 
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Here is a real application of probability. This relates to the concept of a LOTTERY, a type of game offered in many states. In some forms of LOTTO (as it is called), a player pays $1 to pick 6 numbers from the set {1,2,3,...,47,48,49}. On a specified date, the state lottery commission then randomly picks six numbers from the set. If your six numbers match the six chosen by the lottery commission, you win millions of dollars. We can ask: What is the probability that you will win if you play one game?
You now have the MATH POWER to realize that the probability is (_{6}C_{6})(_{43}C_{0})/(_{49}C_{6}) = 1/(13,983,816). In other words, you have about 1 chance in 14 million of winning.
OK, let's assume you play one game a day. We can ask questions like: How many days would you expect to have to play in order to have a 1% chance of winning at least once? (Read the question carefully!) If we let x = the number of days, then we answer the question by solving
1  [(13,983,815)/(13,983,816)]^{x} = .01
==> [(13,983,815)/(13,983,816)]^{x} = .99
==> x = log(.99)/log [(13,983,815)/(13,983,816)]
==> x = 140,542 (days), which is about 385 years.
In other words, if you could play once a day for 385 years, you would be investing $140,542 to have a 1% chance of winning the millions that LOTTO has to offer. Not a good "investment." However, do bear in mind that people do win at LOTTO. If millions of people play, it is not mathematically surprising that someone wins every now and then.
OH, THE AMAZING POWER OF MATHEMATICS!
Problem: Consider the number sets S = {1,2,3,4,5} and T = {1,2,3,4,5}. If you randomly choose two numbers from S, and then I randomly choose two numbers from T, what is the probability that we have the same set of two numbers?
Solution (with communication): The number of sets of two numbers that can be chosen from S is _{5}C_{2 }= 10. Since set T is identical to set S, the desired probability is 1/10, or 10%. 
Problem: I have a 12% chance of winning a specific game. How many games would I have to play to have a 95% chance of winning at least one game?
Solution (with communication): The probability of losing x consecutive games is (0.88)^{x}. Hence, 1  (0.88)^{x} is the probability of winning at least one game if you play x games. Hence we want to solve
1  (0.88)^{x} = 0.95
==> (0.88)^{x} = 0.05
==> x = log(0.05)/log(0.88) = 23.4347.
If you play 24 games, you have a 95% chance of winning at least one game. 
