Suppose you are given a cube, you break the cube in 3X3X3 pieces. However, before doing so, all 6 faces of the cube were colored green. All the other faces not colored green will be white after breaking it up. You randomly pick a small cube and see one face which is green. What is the probability that the cube is an edge cube?
You have a total of 4 investments to make of 100 dollars. You can either invest in a stock which if it goes up in value, you gain an additional 100 dollars, if it goes down, you lose 50 dollars. Or you can bet on whether the stock will go up or down at the casino, where you either win 100 if you're right or lose everything if not. Can you devise a strategy where you will always make a profit?
A game where two players each throw two dice each. If any one die matches the other, you win a dollar. If there is no match, you lose a dollar. Would you play this game or not and why?
1. How do you simulate stock price with VBA? 2. Std.=? 3. VaR=? 4. Three doors, it takes you 1 minute to go through the first door, it also takes you 1 minute to go through the second door, it make your speed slow down 1/2 after going through third door. Only the first door is the right way to go out. You choose each door with the equal probability. How much time will you go out in expectation? 5. You have n 1-meter long ropes, and you could connect two nodes together n-1 times. What is the probability for you to finally get a whole n-meter long rope without rings?
Expected length of the longest segment of a unit-length stick broken in 2 places.
What is the probability that three random points on a unit circle would form a triangle that includes the center of the unit circle?
Let's play another game: There are 100 balls, you and I pick the balls in turn, for each round you can pick 2^k balls(k=0,1,2,...). Do you have a strategy to win?
There are N(sufficiently large) numbers that may repeat and x repeats more than N/2 times. Try to find an algorithm with time O(N) and space O(1) to find x.
Some sample questions: Q. You have a biased coin in which the probability of getting a head is 51%. Create an event out of this coin which has a probability of 25%. Q. There is a calculator in which all digits (0 to 9) and the basic arithmetic operators (+, -, *, /) are disabled. However other scientific functions are operational like exp, log, sin, cos, arc tan, etc. The calculator currently displays a 0. Convert this first to 2 and then to 3. Q. There are five points in an equilateral triangle of side length 1. Prove that there are two points with distance less than 1/2.
Build a tree where the parent can have any number of children. Find the coefficient of a quadratic polynomial if you know the value of y when you plug in any x.
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