Probability Puzzles

Q: You are given two urns and 5 black and 5 white balls. An urn will be picked at random and a ball drawn from it. If the ball is white you win. How would you distribute the balls so as to maximize the chances of winning? The Moscow Puzzles: 359 Mathematical Recreations (Dover Recreational Math) A: Put in 1 white ball in one urn and the remaining 4 and the other 5 black balls in the other urn. This maximizes your chances of winning. If the urn with the 1 white ball is chosen you win with 100% probability, if the other urn is chosen you win with probability = \(\frac{4}{9}\). This adds up to $$ P(win) = \frac{1}{2}\times 1 + \frac{1}{2}\times\frac{4}{9} = \frac{13}{18} $$ You can extend this recursively just for kicks. For example if every urn had two urns inside them and you had to fill them up with the balls to maximize you chances how would you do it? It is the same logic, one white ball in each and the last one should have all the remaining (shown in figure) In this case yo...

Q: Sleeping Beauty (SB) has volunteered for a test. The test is structured as follows. On a Sunday SB is put to sleep and after that a coin is tossed. If the coin Comes up HEADS, SB is awoken on the following day (Monday) and asked her belief in the probability that the toss came up heads. The experiment ends. Comes up TAILS, same as above except that at the end of the question her memory of the waking event and question is wiped out and she is awoken the next day (Tuesday) and asked the same question. The experiment ends. The question is, when SB is awoken, what is SB's belief in the probability that it was heads. Note: At any given time, SB has no way to tell if it is a Monday or a Tuesday. A:  While it is tempting (and there are some arguments out there) to say the probability is \(\frac{1}{2}\), the reality isn't so. Here is why. Take a look at the following table. The "X" indicates the situations when SB wakes up. Given that SB does not know which day of ...

The following is one of the most fascinating results from probability and if you are not familiar with it already, it is definitely counter intuitive. So here goes... Q. There are 10 people in a room. What is the probability that two of them share the same birthday? An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition A.There are several explanations on the web for this classic puzzle. If there are exactly two people, the probability they have the same birthday is \(\frac{1}{365}\). More importantly the probability that they will have different birthdays is \(1 - \frac{1}{365} = \frac{364}{365}\). For three people in a room, that works out to \(\frac{364}{365}\times \frac{363}{365}\). Thus the probability that all n people will all have different birthdays is The probability for n >= 365 is fixed to 1 by the pigeon hole principle. It is interesting to see how the above expression behaves by rearranging some of the terms. The above equation can...

Q: There is an urn with a pair each of 3 different coloured balls. You dip in and pick three . What is the probability that you will end up with two balls of the same colour? --> A: There is an easier way than to map out various combinations to crack this one. So long as the pair of balls. Notice, all we want are two of the same colour and we are picking three. Two balls will always be of the same colour so long as all three are not from different colours. The probability that all three are of a different colour is (1)*(4/5)*(1/2) = 2/5. Thus the sort after probability is 1-2/5 = 3/5=60%. Some good books to learn probability Fifty Challenging Problems in Probability with Solutions (Dover Books on Mathematics) This book is a great compilation that covers quite a bit of puzzles. What I like about these puzzles are that they are all tractable and don't require too much advanced mathematics to solve. Introduction to Algorithms This is a book on algorithms, some ...