## Hard puzzles

Given the set of numbers from 1 to n: { 1, 2, 3 .. n } We draw n numbers randomly (with uniform distribution) from this set (with replacement). What is the expected number of distinct values that we would draw?

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Source: CSEblog
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What is the expected number of cards that need to be turned over in a regular 52-card deck in order to see the first ace?

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x & y are two random points selected uniformly between 0 & 1. Using them, create a point uniformly random in circle of radius 1. (uniform means that the probability density is constant)

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Source: Quant Interview
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N points are chosen at random on the cicumference of a circle. A convex n-gon (n sided polygon) is drawn by joining these n points. What is the probability that the center of circle lies inside the region of n-gon?

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Source: a nice senior
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On pressing a button, a random number is generated uniformly between 0 & 1. You keep on generating these numbers until their sum exceeds 1. What is the probability that you need to press the button more than n times? What is the expected number of times you need to press the button?

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