My script isn't working, can someone please! help me? Further explanation in thread.

function M = initializeBoard(N, p)
for i = 1 : N
    for j = 1 : N
        x = rand(1);
        if (x < p)
            M(i,j) = 1;
        else
            M(i,j) = 0;
        end
    end
end
end

Imagine that this functions creates a N x N matrix A (M).

This matrix consists of 1's and 0's, and the distribution of these depends on p, as can be seen:

if (x < p)
    M(i,j) = 1;
else
    M(i,j) = 0;
end

function pathFound = findPath(N, p)

A = initializeBoard(N, p);

scanPath = bwlabel(A,4);
[rowNum, ~] = size(scanPath);
stats = regionprops(scanPath, 'PixelList');

pathFound = false;

for iblob = 1:length(stats)
    if ((min(stats(iblob).PixelList(:,2)) == 1) && (max(stats(iblob).PixelList(:,2)) == rowNum))
        scanPath(scanPath == iblob) = 10*iblob;
        pathFound = true;
    else
        pathFound = false;
    end
end

end

This function should check if there's a path between bottom and top of the matrix, and if there is, return a 'true' value.

clear; close all; clc;

pSteps = 20;
M = 100;
N = 200;

probs = [];
ps = [];

for i = 1:pSteps
p = (1/pSteps) * i;
fprintf('p is at %.2f\n', p);

paths = 0;

for m = 1:M
    
    if findPath(N, p) == 1
        paths = paths + 1;
    end
    
end

prob = paths / 50;

probs = [probs, prob];
ps = [ps, p];
end

plot(ps, probs)

The script above checks for 20 equally distributed values between 0 and 1 what the probability is of finding a path in a matrix with size N x N.

The output I get from N = 100, M = 100, pStep = 20 (0.05 - 1.00):

The output I get from N = 200, M = 100, pStep = 20 (0.05 - 1.00):

However, the output I get from N = 200 and N = 100 is very different from the output I am supposed to get:

My Question: Does anyone know where I made a mistake / why do my graphs not look like the one shown above at all?

[darova]

Where are original equations/formulas?

[The Legend]

Uhh, I don't think there are equations?

If p is very small, matrix A will contain only a few 1s. If p is close to 1 almost all elements will be 1. These two situations are the two phases.