# Counter-Intuitive Problems

In this project, we discussed “Counter-Intuitive” Problems, where the real result is the opposite of the expected result.

The three problems we talked about and tested were:

1. The rope around the world problem- How many feet of rope is required to wrap around the world, and how much more would we need if we raised the rope ONE foot off of the earth’s surface? Upon our calculations, the answer turned out to be a mere 2pi feet, or 6.28 feet, which is much less than people would expect.

2. You are on the Monty Hall game show as a contestant. Monty, the host, calls you down and you play this game. In the game, there are 3 doors. Behind one is the car of your dreams (you name it) and behind the other two are goats. You have to choose one of the three doors, whatever one you think has the car behind it. Once you choose, Monty then reveals one of the other two remaining doors that has a goat behind it, leaving you now with two doors instead. He then asks you if you wish to keep the door you chose. Or to switch your choice to the other door. So, what do you do?

The answer to this, believe it or not, is to switch doors EVERY TIME. The math is in the slideshow linked to this post.

3. Finally, we explored and proved the birthday paradox. Basically, the birthday paradox asks this:

Theoretically, how many people can you put in the same room to find a pair of people with the same birthday?

The answer is 23 people. (Well, sort of)

We used permutations and probability to prove that in a room of 23 people, the liklihood of 2 people sharing the same birthday is 50%, and that percentage only raises exponentially as more and more people enter the room. For example, we experimented with 40 people, and the percentage rose to around 80%.https://docs.google.com/presentation/d/1ve4i7-Mh5iK4d-mUPo6KPs8xoAP5t1NeBLqUmZA8h3A/edit#slide=id.g1447889c57_1_0