urn

A fast multivariate hypergeometric calculator with an intuitive language interface.

Find the probability of drawing a target set of objects from a collection, either with or without replacement.

Display the results as a table or a plot in your terminal.

Using the calculator

The urn program can be run as a shell:

$ urn
urn>

Computations are described in the following form:

PROBABILITY DRAW [number of things]
FROM [collection]
WHERE [zero or more constraints on draw];

To see the total count of possible draws, replace PROBABILITY with COUNT.

By default, the computation assumes that draws are made without replacement. This can be changed by specifying DRAW [number of things] WITH REPLACEMENT.

Let's look at some examples.

Suppose we want to draw without replacement from an urn containing coloured marbles. We want to see the probability that we see at least 2 red and at most 5 blue:

urn> probability draw from red = 5, blue = 7, green = 3 where red >= 2 and blue <= 5;

This returns the table of probabilities:

  draw size    probability
-----------  -------------
          2      0.0952381
          3      0.241758
          4      0.406593
          5      0.566434
          6      0.706294
          7      0.818182
          8      0.888889
          9      0.895105
         10      0.818182
         11      0.661538
         12      0.446154
         13      0.2

Note that the query keywords such as FROM and WHERE are not case sensitive. A semicolon ; ends the query. Whitespace is ignored.

We didn't specify a size for our draw, so the program returned all draw sizes with a non-zero probability of meeting our constraints.

By default urn returns float numbers for probabilities. We can make it show exact rational numbers by appending show rational:

urn> probability draw 1..5 from red=5, blue=7, green=3 where red >= 2 and blue <= 5 show rational;
  draw size  probability
-----------  -------------
          1  0
          2  2/21
          3  22/91
          4  37/91
          5  81/143

Here we also specified a range 1..5 for the draw size. Single draw sizes (e.g. 5) are also permitted.

It's useful to create a plot (show plot) to see the optimal draw size at a glance:

urn> probability draw from red=5, blue=7, green=3 where red >= 2 and blue <= 5 show plot;
┌────────────────────────────────────────────────────────────┐
│                                ▝     ▘                     │ 
│                           ▘               ▝                │ 
│                                                            │ 
│                     ▗                                      │ 
│                                                 ▘          │ 
│                                                            │ 
│                ▘                                           │ 
│                                                            │ 0.5
│                                                      ▖     │ 
│          ▝                                                 │ 
│                                                            │ 
│                                                            │ 
│     ▝                                                      │ 
│                                                           ▝│ 
│                                                            │ 
│▘                                                           │ 
│▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁│ 0.0
└────────────────────────────────────────────────────────────┘
   2             5             7           10            12

To see the same calculation, but in the case where we draw with replacement, we must specify a range and use the WITH REPLACEMENT modifier:

urn> probability draw 2..13 with replacement
...  from red=5, blue=7, green=3
...  where red >= 2 and blue <= 5 show plot;
┌────────────────────────────────────────────────────────────┐
│                           ▖    ▝     ▖                     │ 
│                                                            │ 
│                     ▗                     ▝                │ 
│                                                            │ 
│                                                 ▘          │ 
│                ▘                                           │ 
│                                                      ▖     │ 0.5
│                                                            │ 
│          ▝                                                ▗│ 
│                                                            │ 
│                                                            │ 
│     ▝                                                      │ 
│                                                            │ 
│                                                            │ 
│▖                                                           │ 
│                                                            │ 
│▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁│ 0.0
└────────────────────────────────────────────────────────────┘
   2             5             7           10            12

Finally, we can use OR to specify any number of alternative constraints on our draw.

urn> probability draw 1..10 from red=5, blue=7, green=3
...  where red >= 2 and blue <= 3
...  or blue > 0 and green > 1
...  or blue = 2 red >= 2 and green <= 2
...  show plot;
┌────────────────────────────────────────────────────────────┐
│                                       ▗      ▝      ▘     ▝│ 1.0
│                                                            │ 
│                                 ▘                          │ 
│                    ▖                                       │ 
│                          ▗                                 │ 
│             ▖                                              │ 
│                                                            │ 
│                                                            │ 
│                                                            │ 0.5
│                                                            │ 
│                                                            │ 
│      ▗                                                     │ 
│                                                            │ 
│                                                            │ 
│                                                            │ 
│                                                            │ 
│▖▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁│ 0.0
└────────────────────────────────────────────────────────────┘
       2             4            6            8           10

To exit the shell, type quit:

urn> quit;
Exiting urn.