The Monte Hall Problem

One of the fun probability mind-bogglers of our time is that of a game show. Imagine you are on a game show and you are presented 3 curtains. Behind one is a brand new car, behind the others are goats. Obviously you'd want the car. So you chose one of the three curtains. Now obviously the probability you chose right is 1/3 or  P(1/3). But if this game show is anything like Monte Hall's, after you chose your curtain they reveal one of the curtains WITH a goat behind it. Now they give you the option to switch. So the question is do you have a higher chance of winning the car if you switch, or if you stick to your guns.

Some people will say it doesn't matter, it just becomes a 50-50 chance right? Well not exactly. So lets assume that you a a contestant will never switch. If you pick curtain #1, and then Monte Hall (your game show host) shows that curtain number three has a goat too, you still have the same chance of winning even if  he hadn't shown the curtain, 1/3. Now if a mathematician comes on to the show, this is how he would approach it. The probability of getting the first guess wrong, is 2/3, and if you get the first guess wrong (choose a goat) and then switch, you will always get the car. This is because obviously you're not going to choose the revealed goat curtain. So, you want to get a goat on the first guess, so you can see where the other goat is, and then switch to the car. The probability of getting the car if you always switch becomes 2/3.

So in conclusion, always switch on those game shows. Your probability increases from 1/3 to 2/3's if you do.

The Spread of Info: Exponential Functions

If you start with one person knowing a rumor and if that person tells 3 people every hour, and if every person that knows this rumor also tells 3 people per hour, 14,348,907 people will know after just 15 hours.

The spread of information is clearly an exponential growth, and the model we are using doesn't take into account many other factors that play a part in how many people know the rumor. However the number we end up with after 15 hours using this very general model is unbelievable.

If we used a more realistic model where 10 new people found out from every person that knew every hour, in just 2 hours, 100 people know. Although, even this is conservative, realistically 10 people find out every 15 seconds. If we graph these exponential functions they quickly result in numbers that are simply too large to understand. It is important to realize that in this day and age, information spreads quicker than ever and often this information can be hurtful, even life-ruining in the case of Monica Lewinsky.

Some Math Humour

New Year calls for some fun... math style.



https://www.buzzfeed.com/generalelectric/25-jokes-only-math-geeks-will-understand?utm_term=.vcPyARmQO#.qoWd1e8Lb



https://www.pinterest.com/pin/179369997635039136/



https://www.pinterest.com/pin/27232772719998822/



https://behappy.me/canvas/why-was-the-math-textbook-sad-because-it-had-problems-then-we-solved-them-210445

I put my root beer in a square glass. Now it’s just beer.

After a talking sheepdog gets all the sheep in the pen, he reports back to the farmer: “All 40 accounted for.”

“But I only have 36 sheep,” says the farmer.

“I know,” says the sheepdog. “But I rounded them up.”

http://www.rd.com/jokes/math/

Why did I divide sin by tan? Just cos.

http://www.jokes4us.com/miscellaneousjokes/mathjokes/

http://www.doodlecraftblog.com/2013/03/national-pi-day-funny-math-geek-shirt.html#

https://www.pinterest.com/pin/23855073003068699/



https://www.pinterest.com/pin/419960733980815975/



https://www.pinterest.com/pin/555279829030764608/

That's all. I hope you laughed :)

Proving Trig Identities

Proving Trig Identities

An identity, is an equation that is true for all "X" values.  Calculators can help us prove when an equation IS NOT an identity. However calculators cannot prove if an equation IS an identity. This is because a graphing calculator cannot display all values of "X" at one time. The display window in restricted to a finite number of values. This is problematic because if there is even one point where the two lines aren't the same outside the view able window, it wouldn't be an identity. To prove my above-stated point, I will prove that a trig equation isn't an identity, and show that a trig identity can't be proven with a calculator, but algebraically.

In sum:
a. Find an equation that is not an identity.  Use your calculator to demonstrate that it's not an identity. Be creative. Take screen shots, pictures, images, etc.b. Find an identity.  Use your calculator to show what an identity looks like on the screen. Be creative. Take screen shots, pictures, images,etc.c. Why can you not prove an identity using the graphing or tables tools? Explain.

A. I think we can all agree that tanx/secx=csx/tanx isn't true. It would be an identity if all values of x were the same for both graphs were the same(the lines would be the same). 

Obviously this isn't an identity and this can be proved by the calculator.

B. Let's take the equation (cot^2x)7 = (1 + cosx^2)7 



















so while this looks like an identity, we cannot prove it with just this image. What if there where an inconsistency outside of the view-able frame? Then it wouldn't be an identity. So, we have to prove it algebraically. This is done as such:

Trigonometry and houses

Without Trig, your house would be lopsided.

Architects and contractors use trig to find the "pitch" of a roof while road constructors use the term "grade". To understand "pitch" and "slope" we must understand the trigonometric function tangent. When talking about the pitch of a roof, a fraction is used to describe it. For instance 11/18. For every 11 inches rise, there are 18 inches of run. Rise/Run or Opposite/Adjacent is the notation for tangent as well!


So if the full length of the "run" is 22 feet, lets calculate the full length of the rise, and then the angle of elevation.
11/18 = x/22, x = 36.
So if the angle of elevation is Rise/Run, then 36/22 is the slope! The tangent function works such, that if you plug in the angle of elevation, that we would get the slope. The inverse of the tangent function however will provide the angle if you provide the slope. So to find the angle of elevation, we plug our slope into the inverse tangent function.
tan^-1(36/22) = 58.6 degrees
reference:
http://jwilson.coe.uga.edu/EMAT6680Su09/King/Roofing/Application%20of%20%20Mathematics%20in%20Construction.htm

I was moved by Mr. Hoffman talking about all the interesting women he had never met because of their appearance. I don't think this only applies to people that one would be sexually attracted to, but to all. Not every person will be interesting to you, but to give everyone a chance to make an impression on you is important. We ask ourselves why we aren't more open, kind people. Why we don't judge someone by their appearance. George Syracuse answers this. We are self interested entities. And as we grow older we regret the missed opportunities to learn a person's story and bond with them. This "brainwash" that Hoffman talks about is all too apparent in present society especially in high school. It's frustrating. To see a judgement made about someone before even interacting with them. I think it is this frustration that should drive us to be kinder people.