Q1. Three points #1, #2, and #3 are selected at random from the circumference of

Question

Q1. Three points #1, #2, and #3 are selected at random from the circumference of a circle (see figure). Write Matlab code to find the probability that the three points lie on the same quarant. Plot the result. Notice: The solution for each answer should be the Matlab code, followed by the plots. Answer which has no Matlab code will not get any credit. 90 degree Q2.Use Matlab to calculate the probability to get 3 times of "5" in 8 throws of a fair die. Q3. The probability to have exact 3 boys in a 5 kids family, if 1. The birth rate of boy is equal to that of girl 2. If the birth rate of boy is 1/3 and girl is 2/3. Please find the answer by using Matlab simulation

Explanation / Answer

function prob = sumDicePDF(n,k)
input: 1. number of dices(n >= 1) 2. k - integer number from n to 6n
output: probability
vec = ones(1,n - 1);
sumOfCounting = 0;
for i = 1:6
    for j = 1:length(vec)
       vec(j) = i;
       sumOfvalue = sum(vec);
       if k - sumOfvalue < 7 && k - sumOfvalue > 0
           for m = 1:length(vec)
               sumOfCounting = sumOfCounting + length(vec) + 1 - m;
           end
       end
    end
end
or
% Roll the dice "numberOfRolls" times
numberOfRolls = 200; % Number of times you roll all 6 dice.
n = 10; % Number of dice.
maxFaceValue = 6;
rolls = randi(maxFaceValue, n, numberOfRolls)
% Sum up dice values for each roll.
columnSums = sum(rolls, 1)
% Find out how many times each sum occurred.
edges = min(columnSums):max(columnSums)
counts = histc(columnSums, edges)
% Normalize
grandTotalSum = sum(counts(:))
normalizedCountSums = counts / grandTotalSum
bar(edges, normalizedCountSums, 'BarWidth', 1);
grid on;
% Enlarge figure to full screen.
set(gcf, 'units','normalized','outerposition',[0 0 1 1]);
% Give a name to the title bar.
set(gcf,'name','Demo by ImageAnalyst','numbertitle','off')
title('Frequency of Roll Sums', 'FontSize', 40);
or
% function: sumDicePDF
    % purpose: Calculate the probability function of the sum of dice
    %           Given n fair dice. Let define X(i) the value
    %           out in the cube (i).
    %           Define Y(n) = sum(X(i)).
    %           Calculate the probability function of Y(n).
    %           Means the P(Y(n) = k)
    %           for given k.
    % input: n - number of dices. n >= 1 (integer)
    %        k - integer betwen n and 6n
    % output: probability - P(Y(n) = k)
    % example: sumDicePDF(3,4) ans = 0.0139
    %          sumDicePDF(8,20) ans = 0.0218
function prob=sumDicePDF(n,k)
    %Check the input
    if n < 1 || k < n || k > 6*n
        display('Bad Input! Try Again.');
        n = input('n = ');
        k = input('k = ');
        prob=sumDicePDF(n,k);
        return;
    end
    %If there is only one dice return probability 1/6
    if n == 1
        prob = 1/6;
        return;
    end
      %We will use equalition: f(n,k)=sigma(i = 1:6)[f(n-1,k-i)*1/6]. The
      %idea is to create matrix (f(1,1) f(1,2) ... f(1,6n))
      %                         (...... ...... ... .......)
      %                         (f(n,1) f(n,2) ... f(n,6n))
      %And use it to find out f(n,k) that we need with equalition
      valuesMatix = zeros(n - 1,6*n); %Define new matrix size n-1,n*6
                                       %with zeros
      %Fill the first row for 1<x<6 with 1/6 and the rest leave with zeros
      for i = 1:6
          valuesMatix(1,i) = 1/6;
      end
      %Fill the rest of matrix with relivant values
      for i = 2:n - 1
          for j = i:6*i             
              probForSpecPlace = 0;
              for l = 1:6
                  if j - l > 0 %Check if value out of range becouse
                               %(k - i) must be positive
                      %Sum the Probabilites for specific f(n,k)
                      probForSpecPlace = probForSpecPlace + valuesMatix(i - 1,j - l)*1/6;
                  end
              end
              %Put that value in its place in matrix for futher use
              valuesMatix(i,j) = probForSpecPlace;             
          end
      end
      prob = 0; %Define the probability to 0
      %The last module: find the exact value for wanted probability using
      %the equalition
      %by using the created matrix of values
      for i = 1:6
          if k - i > 0
              prob = prob + valuesMatix(n - 1,k - i)*1/6;
          end
      end
end

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