Assignment 80
"Without the concepts, methods and results developed by previous methods right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years."  (Hermann Weyl, 18851955)
Srinivasa Ramanujan (18871920): Born in India and primarily selfeducated, he produced amazing mathematical results that, when discovered, boggled the minds of modern mathematicians. Incredibly, his work was not well known until 1976 when 130 pages of his scribbled material were discovered in a box of letters and bills in the library of Trinity College in Cambridge. Since then mathematicians have been studying these notes (known as the Lost Notebook ) and hundreds of other handwritten pages of Ramanujan's notes found after his death. Ramanujan had jotted down thousands of formulas, almost always without proof or even a hint of where they came from. Some of Ramanujan's assertions have yet to be proved, but most have been established as valid and useful in modern mathematics, including computer technology. (The main character in the movie Good Will Hunting is patterned after Ramanujan, and Ramanujan's name is mentioned multiple times in the film.) 
Why did Herkimer put a pill under his pillow?
Answer: Someone told him it was a cold pill.
Herky wants to know:
Do you insult a person who dismantles roofs if you call him an eavesdropper?
If you are pretending to be sick while it is raining, are you being untruthful if you say that you are "under the weather?" 
ASSIGNMENT #80
Reading: Section 12.6, pages 739742.
Written: These two problems (both 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: SUM OF SQUARES NUMBER: A number that is the sum of the squares of consecutive integers, starting with 1. The first four sum of squares numbers are 1^{2} = 1, 1^{2}+2^{2} = 5, 1^{2}+2^{2}+3^{2} = 14, 1^{2}+2^{2}+3^{2}+4^{2 }= 30.  Requirements for a binomial setting:
(1) There are N independent trials; (2) Each trial has only two possibilities... success or failure; (3) The probability of success for each trial is the same.
Here is an example:
Assume that there is a 30% probability that I will win a particular game. If I play five games, what is the probability that I will win x games, where x assumes the values 0,1,2,3,4,5?
Note: This is a binomial setting. We have N = 5, each game is independent of any other game, and the probability of success is p = 0.3, or 30%. The probability of failure is 1  p = 1  0.3 = 0.7 = 70%.
The probability I will win x games is _{5}C_{x}(.3)^{x}(.7)^{5x}.
Remember... the variable x can assume the values 0,1,2,3,4,5.
Here is a probability chart.
Value of x 
Probability 
0 
_{5}C_{0}(.3)^{0}(.7)^{5} = 0.16807 
1 
_{5}C_{1}(.3)^{1}(.7)^{4} = 0.36015 
2 
_{5}C_{2}(.3)^{2}(.7)^{5} = 0.30870 
3 
_{5}C_{3}(.3)^{3}(.7)^{2} = 0.13230 
4 
_{5}C_{4}(.3)^{4}(.7)^{1} = 0.02835 
5 
_{5}C_{5}(.3)^{5}(.7)^{0} = 0.00243 

Total = 1.0000 
Problem: In a specific game, I have a 20% chance of winning. If I play two games, find the probability that I (a) win both games; (b) win exactly one game; (c) lose both games.
Solution (with communication):
(a) _{2}C_{2}(.2)^{2}(.8)^{0 }= 0.04 = 4%.
(b) _{2}C_{1}(.2)^{1}(.8)^{1} = 0.32 = 32%.
(c) _{2}C_{0}(.2)^{0}(.8)^{2} = 0.64 = 64%.
Note: The sum of the probabilities is 100%, as should be expected. 
Problem: In an given trial, I have a 98% chance of success. If I conduct 100 trials, what is the probability that I will have no more than one failure?
Solution (with communication): The requested answer is the probability of no failures plus the probability of exactly one failure. This is
_{100}C_{0}(.02)^{0}(.98)^{100 }+ _{100}C_{1}(.02)^{1}(.98)^{99}
= 0.1326196 + 0.2706522
= 0.4032718, or about 40.3%. 
