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* Two bullets are put into a gun's round barrel consecutively, which has capacity of 6. The gun is shot once, but no bullet is fired. Does rolling the barrel (shuffling) before next shot increase the probability of firing a bullet?
* A horn is formed by rotating 1/x around the x-axis from 1 to infinity. What's the probability that a random point inside the horn has x-coordinate less than 2?
* Estimate the number of piano tuners in Chicago if pianos can tune themselves using AI. Provide a 95% confidence interval.
* We have a weighted coin which shows a head with probability p, (0.5<p<1). How do we get a fair toss from this? That is, how do we toss this coin in such a way that we can have probability of winning = loosing = 50%?
* We have a weighted coin which shows a head with probability p, (0.5<p<1). How do we get a fair toss from this? That is, how do we toss this coin in such a way that we can have probability of winning = losing = 50%?
* Calculate the VaR of a portfolio whose returns follow a nowhere-differentiable function (e.g., Weierstrass curve) with Hausdorff dimension 1.5.
* You’re on a game show with countably infinite doors. Behind one is a car; the rest have goats. After you pick a door, the host (who knows what’s behind all doors) opens all but one other door, revealing goats. Should you switch?
* There is a regular die and a special invisible die. You know that regular die has integers 1 to 6, but don't know what's on the invisible dice. After tossing, I speak the sum of outcome of both die. It so happens that the outcome is an integer between 1 to 12, with equal probability (1/12 each). Can you guess what are the numbers printed on special invisible dice?