Simulating finding a randomly chosen number by asking binary questions

As a question in an assignment, I have been asked to write an Octave function that simulates 1000 experiments of finding a Random Variable X with alphabet {0, 1, 2, 3} and pmf:

Px(0) = 1/8

Px(1) = 1/4

Px(2) = 1/2

Px(3) = 1/8

by asking a sequence of binary, "yes" or "no" questions.

I have determined that the optimal sequence of binary questions asked to find the value of X is to simply ask "Is X = p?" where p are the possible values, in order of decreasing probabilty.

So the optimal sequence would be:

1. Is X = 2?

   If not:

2. Is X = 1?

   If not:

3. Is X = 0?

   If not then X = 3

This is the function I have written:

function x = guessing_experiment(probabilities, n)
  % generates n simulations of finding a random number in an alphabet by asking binary questions,
  % where 'probabilities' is a list of the probabilities per number in the order the questions will be asked

  num_Qs = zeros(1,n);                            % allocate array of size n for number of questions asked per experiment
  [num_col, alphabet_size] = size(probabilities); % get size of alphabet

  for i = 1:n                                     % generate n experiments
    Qs = 0;                                       % number of questions asked in this experiment
    for j = 1:alphabet_size - 1                   % iterate through questions
      question = rand;                            % generate random number in range [0, 1]
      Qs++;                                       % incremenet number of questions asked
      if (question <= probabilities(j))           % if question produces a "yes" answer
        break;
      endif
    endfor
    num_Qs(i) = Qs;                               % store number of questions asked for this experiment
  endfor

  x = mean(num_Qs);                               % calculate mean number of questions asked over the n experiments 

 end

Which is called as guessing_experiment([1/2, 1/4, 1/8, 1/8], 1000) where the array is the probability of each question producing a "yes" answer, in order of how they are to be asked, and n is the number of experiments.

Asking these questions should produce an average number of questions of 1.75, but my program is always producing an average of ~1.87. Where is my script in error?