Monday, February 11, 2013

Urns and Balls Puzzle

Q: There are a large number of urns with each urn having as many balls as there are urns. There is exactly 1 red ball within each urn and the rest are white. You are allowed to draw exactly one ball from each of the urns in sequence. If you ever get a red ball, you win a prize. What is the probability that you will win?

The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (And Everone Else!)

A: Assume there are \(n\) urns. This implies that there are \(n\) balls per urn with exactly 1 red ball and \(n-1\) white balls. The probability that you would draw a red ball from a given urn is \(\frac{1}{n}\) implying the probability that you would not win from a given urn is \(1 - \frac{1}{n}\). The probability of not winning at all is

$$ P(\text{no win}) = (1 - \frac{1}{n})\times (1 - \frac{1}{n})\ldots \text{n times}\\ = (1 - \frac{1}{n})^{n}$$

Note, that for large \(n\), the above has a limit which converges to

$$ \lim_{n\to\infty}(1 - \frac{1}{n})^{n} = \frac{1}{e}$$

This further implies that the probability of winning is \( 1 - P(\text{no win}) = \frac{e - 1}{e} = 63\%\)

If you are looking to learn probability, here are some great books to own.

Introduction to Algorithms

Introduction to Probability Theory



Introduction to Probability, 2nd Edition

The Mathematics of Poker


Quantum Poker




Discovering Statistics Using R

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