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Showing posts from November, 2013

### The Sultan's Wine Bottles

Follow @ProbabilityPuz Q: A Sultan has a 1000 bottles of wine. He needs to use them in 30 days time for a royal banquet. He knows that his enemies have poisoned exactly one bottle with a type of poison that takes effect in 29 days. He decides to use his soldiers to test which bottle is poisoned. Is there a strategy that minimizes the number of soldiers needed for the task? Probability Theory: The Logic of Science A: The naive approach is to have one soldier per bottle. Every soldier gets a drop from each bottle and they wait for 29 days. The number of the soldier who gets affected on the 29th day shows which bottle is poisoned. However, this strategy is quite expensive in terms of the number of soldiers needed for the Sultan. A far more efficient strategy is the following. Label each bottle with a number. Maintain a ledger which maps a number to each of the patterns 0000000000 -> 1, 0000000001 -> 2, 0000000010 -> 3, 0000000100 -> 4 and so on till you rea

### Points on a Circle

Follow @ProbabilityPuz Q: A total of $$n$$ random points are selected from a circle. What is the probability that all of them are on a semicircle when evaluated in a clockwise order? Probability Theory: The Logic of Science A: Assume $$n=4$$ as shown in the figure below. Without loss of generality, assume the points were picked in the order shown by the numbers in the figure. Start with the point numbered 1 and look in a clockwise direction. Each of the subsequent points can be either in or out of the semicircle starting from 1. Each of the given points will be within the semicircle with probability $$\frac{1}{2}$$. Therefore the probability that all the remaining $$n-1$$ points will be in the clockwise semicircle is $$\frac{1}{2^{n-1}}$$. This estimate is for a given starting point numbered at 1. To compute the probability for all $$N$$ points, we apply the same logic to each of the points in sequence. This yields $$\frac{n}{2^{n-1}}$$ which is the probability that