I'm after a nice method to draw from a range of integers given known prior probabilities.

Say I wanted an 80% chance of drawing 1, a 15% chance of drawing 2, and a 5% chance of drawing 3...

I'm obviously thinking about this wrong, since neither

[~,I] = max([.80 .15 .05] .* rand(1,3))


[~,I] = max([.80 .15 .05] + rand(1,3))

appears to achieve this...(rand samples from a uniform distribution)

Does anyone have a suggestion that will extend to any range of integers?

** Additional Comment **

I realise now for rational probabilities it's trivial just to create a vector comprised of these options proportionally (80% 1s, 15% 2s, and 5% 3s), and then just randperm the contents of the vector and choose the first element over and over...

This method is bad if the probabilities do not behave nicely...

  • $\begingroup$ In R, there's the sample and sample.int functions. e.g. sample(3,100,replace=TRUE,p=c(.8,.15,.05)) will generate 100 values from the distribution in your question. $\endgroup$
    – Glen_b
    Aug 13, 2017 at 0:56
  • $\begingroup$ Thanks @Glen_b. I need to implement a solution in MATLAB...and so it would be great to understand how this function in R is achieving this... $\endgroup$
    – NBland
    Aug 13, 2017 at 3:39
  • $\begingroup$ R is open source; you can read the code but I don't think you'll find it as enlightening in this case as the indicated duplicates. The table method would probably serve you well. It's quite fast -- you take a longish array and fill it with values in the right proportions so that you are almost sampling with the required probabilities; there's normally a few "leftover" cells at the end which take you to a second step of generation that take you up to the exact probabilities you need (and that step can use any convenient other method) $\endgroup$
    – Glen_b
    Aug 13, 2017 at 5:39

1 Answer 1


Why don't you split your interval into three pieces of unequal length and assign the corresponding integer. Here the Matlab code:

%% define percentage
perc = [80, 95, 100];

%% number of random integers
nMax = 1e3;

%% generate integers
vec = randi([1, 100],1,nMax);
t   = toc;
disp(['Generated ' num2str(nMax) ' random integers in ' num2str(t) 's'])

%% get indices
% idx(numel(perc)).dat = NaN; % only needed if you have many intervals
for i = 1:numel(perc)
   idxVec(i).dat      = find(vec <= perc(i));
   vec(idxVec(i).dat) = NaN;
t = toc;
disp(['Indexed ' num2str(nMax) ' random integers in ' num2str(t) 's'])

%% plotting:
for i = 1:numel(perc)
    plot(idxVec(i).dat, i*ones(1,numel(idxVec(i).dat)), '.');
    hold on;
hold off

I'm not sure how big your database is. However, for my standard the code is fast:

Generated 10 000 000 random integers in 0.21746s
Indexed 10 000 000 random integers in 0.35026s

PS: You should remove the time taking (tic, toc) and the figure. Especially the plotting will take time for a large database.

  • $\begingroup$ This is reasonable for my given example, but it becomes a bit clunky when I want to choose from a larger range of values. Thanks for your answer. :) $\endgroup$
    – NBland
    Aug 13, 2017 at 3:36
  • $\begingroup$ I edited the code, so that it is optimized for Matlab. $\endgroup$
    – Semoi
    Aug 15, 2017 at 17:34

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