Probability Puzzles

I chanced on this excellent puzzle on the net that tends to reveal a cognitive bias in our heads against Bayesian reasoning. The puzzle statement is quite simple You are given four cards. Each card has a letter on one side and number on the other side. You are told the statement "If there is a D on one side, there is a 5 on the other side". Which two cards would you flip over to validate the statement? The original article is here , think hard before you click through for an answer :)

Follow @ProbabilityPuz Q: You are about to play a game wherein you flip a biased coin. The coin falls heads with probability \(p\) and tails with \(1 - p\) where \(p \le \frac{1}{2}\). You are forced to play by selecting heads so the game is biased against you. For every toss you make, your opponent gets to toss too. The winner of this game is the one who wins the toss the most. You, however get to choose the number of rounds that get played. Can you ever hope to win? Machine Learning: The Art and Science of Algorithms that Make Sense of Data A: At a first look, it might appear that the odds are stacked against you as you are forced to play by choosing heads. You would think that your chances or winning decrease as you play more and more. But, surprisingly there is a way to choose the optimal number of tosses (remember, you get to choose the number of times this game is played). To see how, lets crank out some numbers. If you get to toss the coin \(n\) times, then t...

The following are some of the best books to own to learn Monte Carlo methods for sampling and estimation problems Monte Carlo Methods in Statistics (Springer) This is a good book which discusses both Bayesian methods from a practical point of view as well as theoretical point of view with integrations. The explanations given are also fairly comprehensive. There are also a fair amount of examples in this text. Overall, this is an excellent book to own if you want to understand Monte Carlo sampling methods and algorithms at an intermediate to graduate level. Explorations in Monte Carlo Methods (Undergraduate Texts in Mathematics) This is good book to own to get you started on Monte Carlo methods. It starts with fairly simple and basic examples and illustrations. The mathematics used is also fairly basic. Buy this if you are at an undergraduate level and want to get into using Monte Carlo methods but have only a basic knowledge of statistics and probability. Monte Carlo Methods in...

Follow @ProbabilityPuz Q: You are conducting a survey and want to ask an embarrassing yes/no question to subjects. The subjects wouldn't answer that embarrassing question honestly unless they are guaranteed complete anonymity. How would you conduct the survey? Machine Learning: The Art and Science of Algorithms that Make Sense of Data A: One way to do this is to assign a fair coin to the subject and ask them to toss it in private. If it came out heads then answer the question truthfully else toss the coin a second time and record the result (heads = yes, tails = no). With some simple algebra you can estimate the proportion of users who have answered the question with a yes. Assume total population surveyed is \(X\). Let \(Y\) subjects have answered with a "yes". Let \(p\) be the sort after proportion. The tree diagram below shows the user flow. The total expected number of "yes" responses can be estimated as $$ \frac{pX}{2} +...

Follow @ProbabilityPuz A class of analysis that has piqued the interest of mathematicians across millennia are Diophantine equations . Diophantine equations are polynomials with multiple variables and seek integer solutions. A special case of Diophantine equations is the Pell's equation . The name is a bit of a misnomer as Euler mistakenly attributed it to the mathematician John Pell. The problem seeks integer solutions to the polynomial $$ x^{2} - Dy^{2} = 1 $$ Several ancient mathematicians have attempted to study and find generic solutions to Pell's equation. The best known algorithm is the Chakravala algorithm discovered by Bhaskara circa 1114 AD. Bhaskara implicitly credits Brahmagupta (circa 598 AD) for it initial discovery, though some credit it to Jayadeva too. Several Sanskrit words used to describe the algorithm appear to have changed in the 500 years between the two implying other contributors. The Chakravala technique is simple and implementing it ...

Follow @ProbabilityPuz All too often, when we deal with data the outcome needed is a strategy or an algorithm itself. To arrive at that strategy we may have historic data or some model on how entities in system respond to various situations. In this write up, I'll go over the method of reinforcement learning. The general idea behind reinforcement learning is to come up with a strategy to maximize some measurable goal. For example, if you are modelling a robot that learns to navigate around obstacles, you want the learning process to come back with a strategy that minimizes collisions (say) with other entities in the environment. Pattern Recognition and Machine Learning (Information Science and Statistics) For the sake of simplicity, lets assume the following scenario. A robot is placed (at random) on flat plank of wood which has some sticky glue in the center. To its left there is a hole which damages the robot a bit and to its right is a reward which is its dest...

Follow @ProbabilityPuz Q: You are in a game where you get to toss a pair of coins once. There are two boxes (A & B) holding a pair each. Box A's coins are fair however B's coins are biased with probability of heads being \(0.6\) and \(0.4\) respectively. You are paid for the expected number of heads you will win. Which of the boxes should you pick? Machine Learning: The Art and Science of Algorithms that Make Sense of Data A: The expected number of heads if you chose box A is easy to calculate as $$ E(\text{heads}| A) = \frac{1}{2} + \frac{1}{2} = 1 $$ However the expected number of heads if you chose box B is also the same $$ E(\text{heads}| B) = \frac{4}{10} + \frac{6}{10} = 1 $$ The average yield being the same could make one think that both boxes yield the same. However there is one difference, its the variance. The variance of a distribution of a random variable \(X\) is defined as $$ Var(X) = \sum_{i=0}^{N} (x_i - \bar{x})^{2}p_i $$ where ...