I was taking a calculus competitive test, and I encountered this problem:
Henrik is randomly choosing numbers between $0$ and $1$ until the sum of all of the numbers that he has chosen exceeds $1$. What is the expected number of numbers Henrik will choose?
I spent a good amount of time on this, but I can't even think of a way to approach the problem.
I'm currently taking AP Calculus BC, so I know a good amount of calculus, but I feel like I'm missing the intuition to tackle problems like this where calculus is applied to probability.
Any help would be appreciated.
I'm not sure how the question was intended to be answered but I suspect they wanted you to a formula for the expected sum.
From probability it would be the sum of the expected numbers randomly drawn so n (the number of draws) times the expected (average) number seen on a draw. This is where calculus comes in. If you are randomly drawing numbers from 0 to 1, to find the expected (average) number you draw, you integrate x times probability of drawing x over the range from 0 to 1. With all probabilities equally likely because it is a random draw, this means you integrate x times 1 dx from 0 to 1 to find the average number drawn. Then, take that answer and determine what n is so that n times that average number drawn is greater than 1.