Have you ever wondered why the area of a circle is pi r squared? Today we'll discover this through a fascinating visual proof. We start with a circle of radius r, and we'll slice it into equal sectors like pieces of a pie.
Now comes the magic! We rearrange these sectors alternately, flipping every other one upside down. This creates a shape that looks almost like a rectangle. The width of this rectangle is half the circumference of the circle, which is pi r, and the height is the radius r. Therefore, the area is pi r times r, which equals pi r squared!
Here's a famous probability puzzle that stumps many people. You're on a game show with three doors. Behind one is a car, behind the other two are goats. You pick door 1. The host, who knows what's behind each door, opens door 3 revealing a goat. Now you have a choice: stick with door 1 or switch to door 2. What should you do? The surprising answer is that you should always switch! Your original choice had a 1 in 3 chance of being correct, which means there's a 2 in 3 chance the car is behind one of the other doors. When the host eliminates one wrong door, that 2 in 3 probability transfers entirely to the remaining door.
Here's another counterintuitive probability problem called the Birthday Paradox. In a room of just 23 people, what do you think is the probability that at least two people share the same birthday? Most people guess it's quite low, maybe around 10 or 20 percent. But the actual answer is over 50 percent! With 30 people, the probability jumps to over 70 percent. This happens because we're not asking about a specific birthday match, but any match among all possible pairs of people. The number of possible pairs grows much faster than you might expect.