Math - Cereal Box Problem or Monty Hall Problem

[dessskris]

CEREAL BOX PROBLEM

(http://mste.illinois.edu/reese/cereal/)

Suppose there was one of six prizes inside your favorite box of cereal. Perhaps it's a pen, a plastic movie character, or a picture card How many boxes of cereal would you expect to have to buy, to get all six prizes? It is not practical to go to the store and buy all that cereal at once. Suppose you got together with your friends and after a combined purchase of 8 boxes you do not yet have all six prizes. Should you be surprised? What if you don't have them all after 10, or 15, or 20 boxes? How many boxes do you think that it should take to get all six?

As you do more and more trials, you will find that the number approaches 14.7. This number is given by:

6/1 + 6/2 + 6/3 + 6/4 + 6/5 + 6/6 = 14.7

Similarly, it can be shown that for an eight sided die, the theoretical number (expected value) of rolls needed to get all eightsides is:

8/1 + 8/2 + 8/3 + 8/4 + 8/5 + 8/6 + 8/7 + 8/8 = 21.7

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MONTY HALL PROBLEM

(http://mathworld.wolfram.com/MontyHallProblem.html)

Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do.The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat. But after Monty has eliminated one of the doors for you, you obviously do not improve your chances of winning to better than 1/3 by sticking with your original choice. If you now switch doors, however, there is a 2/3 chance you will win the car (counterintuitive though it seems).

winning probability

pick stick 1/3

pick switch 2/3

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I have decided to do my EE in Math and would not change the subject area. I regret it, though, because it's really hard to find a good topic.

but I came across those two problems and found them interesting. unfortunately the two problems are way too simple that I can't just elaborate any of them in my EE. however my supervisor said that it is possible if I could make the problem more complicated or just find a similar problem which is more complicated than any of these two.

now I would like to ask if any of you guys know any problem like these or any extended version of these problems, any case, any theory, etc

because I have been thinking for months and haven't really come up with a suitable EE topic. even these two problems, I didn't know them before I read them when I was looking for Math problems, which means I do not really follow the Math world, the popular Math problems.

which of the two do you think is better? I also heard the question 'what is the probability of 2 people in a class of 20 (for example) having the same birthday?' but my teacher said it is not worth doing as an EE.

suggestions guys?

thank you