Quote:
Part 1: Your camel's back can hold 1 unit. Straws have weights which are uniformly distributed from 0 to 1 unit, and independent of each other. You add one straw at a time. If the weights are 0.7, 0.1, 0.8..., then your camel's back breaks on the 3rd straw. On average, how many straws does it take to break your camel's back?
There are many solutions. In white, I have written up the one I found without pencil or paper when I was 13 after an uncle posed the problem at a family Thanksgiving dinner, and at which blah_blah hinted.
The expected value of a random variable X taking positive integer values is the sum of the probabilities P(X>=1)+P(X>=2) + P(X>=3) + ... This useful trick is actually a change of the order of summation in something that looks like a single sum, but can be expanded:
E(X)
= P(X=1) + 2 P(X=2) + 3 P(X=3) + ...
= P(X=1) + P(X=2) + P(X=3) + ...
+ _________P(X=2) + P(X=3) + ...
+ _______________ + P(X=3) + ...
Summing rows, this is
P(X>=1)+
P(X>=2)+
P(X>=3)+...
Anyway, P(X>=n) = P(the first n-1 straws add up to less than 1, so with k=n-1 we want the sum of the probabilities that the first k straws add up to less than 1 for k=0,1,... The plane x1+...+xk =1 cuts off a simplex of volumn 1/k! from the unit cube, so the expected value is 1/0! + 1/1! + 1/2! + ... = e.
Also at that dinner, we discussed the geometric mean of the numbers in [0,1], which is again related to e.
I'll post my solutions to Parts 2 and 3 later. I used a method which at least looks different from the suggestions of blah_black and bigpooch. However, I would not be surprised if they were all similar, from some perspective.