These activities are part of Siegrist's Probability/Statistics Object Library project, a wonderful resource for teachers of probability and statistics as well as people interested in developing their own Java applets for educational use. This small sample consists of three classic problems for student exploration: The Birthday Problem, The Poker Problem, and Buffon's Needle Problem. All three are structured for student exploration followed by computer simulations of a particular problem, and each comes with ample exercises to guide students through interesting questions about what they are doing. Some sophistication with notation or interpretation by the instructor will be necessary for younger students. The following is a brief description for each of the three parts:
- The Birthday Problem is the classical unintuitive fact that the probability of two people having the same birthday in a group of 30 is over 0.70. The computer simulation is done with an applet that can be set to perform an experiment hundreds or thousands of times. In this case, the experiment is to pick n numbers from 1 to N and record I = 1 if there is duplication of a number and I = 0 if there is not. (Hence, the situation described above corresponds to n = 30 and N = 365.) This is an intriguing problem for students that can be solved exactly with only a small degree of sophistication -- a terrific introductory problem!
- The Poker Problem allows hundreds or thousands of poker hands to be generated and each tested to find its "value." At the end of the simulation, the proportion of all trials with each possible value (no value, one pair, two pairs, three-of-a-kind, straigh, flush, full house, four-of-a-kind, and straight flush) is displayed numerically and graphically. The computations of the exact probabilities in these cases are often trickier than in the Birthday Problem, but the necessary counting arguments are given for these problems as well.
- Buffon's Needle Problem is a gem from geometric probability in which we find the probability of a needle of length L crossing a line when dropped on a floor with parallel lines 1 unit apart. In this case, the simulation includes an appropriate graphic illustrating the needle and the lines for each trial. In addition to simply counting the number of times that the needle crosses the line, the applet illustrates the position of the needle as a point (x, y) relative to the curves y = (L/2)sin(x) and y = 1 - (L/2)sin(x) so that the standard geometric argument for the exact probability (being 2L / pi) is properly motivated.
Open Three Easy Pieces in a new window
Kyle Siegrist, "Three Easy Pieces - Editorial Review," Convergence (June 2004)