I found this geometric reasoning (or proof) of a derivative of a Tangent function in Tristan Needham’s book on Visual Differential Geometry and it struck me as something really intuitive, simple and beautiful. I wanted to do similar derivations for other simple trigonometric functions, namely, Sine and Cosine of an angle. Although I was able to get the answers, it was not that succint or elegant as the proof of the Tan function....
The Monty Hall problem is an interesting puzzle in probability which has a counter-intuitive answer. In fact, it is documented that many people got the answer wrong and many people continue to get the answer wrong. The problem The problem goes somewhat like this: Imagine a game show. There are 3 doors and behind one door there is a car and behind the other 2, goats. The player picks a door hoping to win a car that’s behind the door....
While I was refreshing some concepts in statistics, I came up with a problem I thought I could ask ChatGPT (talk about attention deficit). I am still using the cheap ChatGPT3.5, so what I am about to show may not work on the expensive ChatGPT4.0. Here’s an example on how it can mislead while solving math problems and one should already have an inkling how to solve the problem (or atleast have an estimate of the answer) to do some prompt engineering to finally get to a solution....
I came across this probability problem and thought it was interesting. What is the expectation of distance (from the center) of a circular disc to uniformly distributed random points on the disc? If you pick a random point (or throw a dart) on a circle, there are more chances of the point landing somewhere between the center and the perimeter of the circular disc. Intuitively, this is because there are more points on the disc, i....