Selected Answers to Problem Set 3
1. I simply checked if June 28, 2030 was a Friday. That's when I turn 50. You can see how I implemented this (everyone's was similar to this one) here.
2. The math approach: if I choose a door at random, it is clear that I have a 33% chance at guessing the right door. Now, if I do not change my guess, then the door I chose still has a 33% chance at being right. If I switch, then that leaves one other door. The door I had guessed at the beginning had a 33% chance of being right and the opened door now has a 0% chance of being right. Therefore, the other unopened door as a 67% chance of being the right door. It is in my best interest to always switch.
Here's a link to a script that I wrote to demonstrate this.
3. The Algebraic Solution: On a stick of length one, the conditions are such:
i) If I assign c to be the longest, then a + b > c ii) if a + b + c = 1, then a + b = 1 - c iii) By substitution, 1 - c > c, or c < 0.5 iv) Therefore, the longest piece must be shorter than 0.5, and all pieces therefore must be shorter than 0.5.Knowing these to be true, for any cut x that I make on the stick, it denotes a range of possible cuts that will yield three pieces which will make a triangle (that is, three pieces all of which are less than 1/2).
i) x < 1/2 ii) y - x < 1/2 implies y < x + 1/2 iii) 1 - y < 1/2 implies y > 1/2 iv) Therefore, 1/2 < y < x + 1/2From this, we can see that if we make a cut x then the odds that y will fall in between 1/2 and (x + 1/2) is exactly x. Of course, we cannot make a cut past one, so when x is past 1/2, y will be between (x - 1/2) and 1/2 (visualize flipping the stick around - if our first cut is past the halfway mark, flip the stick and it's now before the halfway mark). We can instead take the integral of all values of x between 0 and 1/2 and multiply by two (in the picture c1 should be x).
P, then, equals 1/8. Multiplied by two yields the answer, 1/4, or 25%.
Click here for a script to detail this and the two extra credit questions.
4. The expected value is calcualated as (50 * 1/9) + (25 * 3/9) + (10 * 5/9) = 175/9 = 19.44. The standard deviation is calculated as:
SD = sqrt((50 - 175/9)² * 1/9 + (25 - 175/9)² * 3/9 + (10 - 175/9)² * 5/9) = sqrt(103.7380 + 10.2881 + 49.5542) = sqrt(163.5802) = 12.7898
5. Click here for a Perl script demonstrating the Monte Carlo method of finding this area. This script takes a random number between [-3,3] (the x-coordinate), and [-4,4] (the y-coordinate) and sees whether it falls on the ellipse (by checking whether the equation is less than one. If so, it adds one to c. Out of t trials (c/t) we can see the ratio of ellipse area to total area (total area is 6*8, or 48 units). Multipling the ratio, or probability, of the ellipse to the box by the area of the box gives us the approximate area of the ellipse.