1. Let \(DEF\) be an equilateral triangle with side length \(3\) At random, a point \(G\) is chosen inside the triangle. Compute the probability that the length \(DG\) is less than or equal to \(1.\)
2. A stick has a length of \(5\) units. The stick is then broken at two points, chosen at random. What is the probability that all three resulting pieces are longer than \(1\) unit?
1. The probability is pi/9.
2. When we break off the first piece, the probability that the first piece will be longer than 1 unit is 3/5. (This is because we look at the stick as the interval [0,5], then we can break the stick anywhere between 1 and 4.) We are now left with a stick with a length of 4. When we break off the second piece, the probability that the second piece will be longer than 1 is 2/4 = 1/2. This can happen at either end, so the answer is 2*3/5*1/2 = 3/5.