Probability Puzzles

Q: A stick is cut at two points uniformly randomly. What is the probability that the three sticks will make a triangle. A: This problem is best solved by taking a geometrical approach. Without loss of generality, we can assume that the stick has unit length.  Let the cut be made at points A & B. The points A & B can take values uniformly between 0 and 1. The rule for any three segments making a triangle is that the sum of any two sides should be greater than the third. The total number of possible ways can be thought to be the set of points within a unit square as shown below. For any given point (A,B), the stick segments are a, (b-a)  and (1-b), we have three inequalities that need to be simultaneously satisfied (if we assume a < b). The simplification of the above set of equation yields The eq 1 & 2 correspond to the top left quadrant of the unit square, while eq 3 corresponds to the area under the diagonal in that quadrant as shown in the figure be...

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) A good book for graduate level classes: has some practice problems in them which is a good thing. But that doesn't make this book any less of buy for the beginner. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition This is a two volume book and the first volume is what will likely interest a beginner because it covers discrete probability. The book tends to treat probability as a theory on its own Discovering Statistics Using R This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing. Fifty Cha...

Q: What is the average number of cards you need to draw from a well shuffled deck of cards before you get an Ace? The Moscow Puzzles: 359 Mathematical Recreations (Dover Recreational Math) A:There is a lot of discussion on the web on using hypergeometric distributions for solving these kind of problems. The hypergeometric distribution is just a big word for something fairly simple. Here is the wikipedia explanation for it (link) . But an easier "smarter" way to solve this puzzle (along with ones that fit this framework) is to work with expectations and indicator random variables. Indicator random variables are like on/off switches. They are 1 under certain conditions, 0 otherwise. Let us assume that the deck of cards (52 total) is as follows, it has 4 aces and all others are labelled X. Let \(Z_i\) represent the indicator variable for a card in position \(i\) with value set as 1 when all aces are in behind it, 0 otherwise. The total number of draws will then be $$ ...

Q: A & B agree to play a coin tossing game. The coin is fair and has a 50% chance of coming heads or tails. The coin is tossed and the first to flip a heads wins. If A starts the game what is the probability that A would win. A: Assume the probability that A wins the game be \(x\). If A tosses first, she has a probability of \(\frac{1}{2}\) of winning on the first toss. If not, the coin goes to B. When B gets the coin, the probability that B would win is now the same as A. This additionally implies that A would lose with probability \(1 - x\). The value of x can be stated recursively as follows $$x = \frac{1}{2} + \frac{1}{2}\times(1-x)$$ Solving for \(x\) yields $$x = \frac{2}{3}$$ Starting first greatly improves the odds of winning! If you are looking to buy some books in probability here are some of the best books to learn the art of Probability Here are a few Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics) This book is a gr...