A student in the Microelectronics Fabrication Laboratory conducts the following experiment. She creates a square grid on several wafers consisting of N times N equal sized cells. Suppose that after a deposition process that centered on each cell the area that may be conductive is a circle with diameter the same as the length of the side of a cell. Moreover, she finds that each circular area has a probability p of being electrically conducting. She then becomes interested in determining the probability that there is a conducting path from the top to the bottom of the square in each wafer. Of course, a conducting path exists whenever two neighboring cells are touching from bottom to top or from side to side. For example, if N = 2 there are four cells in the square. If we denote a conducting cell with a '1' and and non-conducting cell with a '0' the following two matrices represent two of the cases where there is a conducting path from the top to the bottom: [1 1 0 0] and [0 0 1 1]. (a) If N = 1, what is the sample space of the experiment? What is the probability of the event of interest?
(b) If N = 2, what is the sample space of the experiment? What is the probability of the event of interest, assuming that the event that a cell is conducting is independent of the other cells conducting?
(c) For an arbitrary finite N, what is the cardinality of the sample space?
(d) Develop an algorithm (a Matlab function) that can be used to determine if there is an electrically conducting path from the top to the bottom of the wafer. The algorithm should give a binary output: '0' if there is no conducting path and '1' if there is a conducting path. Test the algorithm when N = 2 and N = 3.
(e) Use the relative frequency definition of probability to estimate the probability that there is a conducting path from the top to the bottom of the wafer for N = 2 when p = 0.1 when N = 10, 20, 50. Repeat for p taking values in {0.5, 0.6}.
(f) Determine for the given values of N if there is a so-called threshold probability p_th such that if p > p_th that the probability that the wafer is conducting is 1. Use a Monte Carlo loop to improve the probability estimate. N = 2: p = 0.5: O = rand (N, N) < p