Amit and Therma meet in a coffee house 5 days in a row. What is the probability that at least once the two will arrive 30 minutes or more apart? > Amit and Therma meet in a coffee house 5 days in a row, and each day they arrive randomly between 2 and 3 (10 arrivals are independent. What is the probability that at least once the two will arrive 30 minutes or more apart? I know I need to break this down into finding the probability that they meat more than 30 mins apart on a daily basis, and only then can I find the probability for 5 days. What is the process for finding what the probability is on one day?

From MathWorld we know that the probability density function for a distance _d_ between two randomly picked points on the unit interval is $2(1-d)$. Here the interval is 1 hour and we require the probability of Amit and Therma arriving more than half that time apart, so compute an integral: $$\int_{\frac12}^1 2(1-x)dx=\frac14$$ This is the one-day probability, and the answer to the original question is straightforward: $1-(\frac34)^5=\frac{781}{1024}$.