Just click the Green Flag A 12 inch ruler is cut at two distinct integer points (not including the endpoints), chosen at random along its length. Meaning the ruler is cut at the 1, 2, 3, ... , 11 inch marks not any fraction of an inch What is the probability that the 3 pieces obtained form these two cuts form a triangle? (Note: Cutting at the endpoints 0 and 12 would just be shaving the ends off the ruler and would not give you 3 pieces. Also, cutting at the same mark twice where Cut 1 = Cut 2 is impossible.) From the pictures you can see only 20 cuts (yellow dots) that make a triangle out of 110 total possible cuts (purple and yellow dots). The purple dots with lines through them are not possible cuts. Thus the probability is 20/110 = 2/11 = .1818 = 18.18% Put the cursor over one of the dots to see the two cuts (x1, x2) and the corresponding side lengths (Side1, Side2, Side3).
Go to my Broken Ruler (Stick) Experiment Simulation https://scratch.mit.edu/projects/91354956/ to see if the Theoretical Probability equals the Experimental after many trials. Note: Have students work this out with different sized measuring sticks (i.e. a yard stick). Use a piece of graph paper to make a picture similar to mine. Also, analyze the continuous case - where you are not just cutting at the integer marks but any fraction in between. For more info, go to my website: https://sites.google.com/site/adamsmathandmore/home/probability
