diff --git a/hail/python/hail/docs/functions/random.rst b/hail/python/hail/docs/functions/random.rst
index 7f61448dad9..babebfd2e85 100644
--- a/hail/python/hail/docs/functions/random.rst
+++ b/hail/python/hail/docs/functions/random.rst
@@ -84,7 +84,7 @@ return the same result, e.g.
      0.6964416913998893,
      0.6092952493383876]
 
-    >>> table = hl.utils.range_table(5, 1).annotate(x=hl.rand_unif(0, 1, seed=0))
+    >>> table = hl.utils.range_table(5, 5).annotate(x=hl.rand_unif(0, 1, seed=0))
     >>> table.x.collect()
     [0.5820244750020055,
      0.33150686392731943,
@@ -92,28 +92,61 @@ return the same result, e.g.
      0.6964416913998893,
      0.6092952493383876]
 
-    >>> table = hl.utils.range_table(5, 5).annotate(x=hl.rand_unif(0, 1, seed=0))
+However, moving it to a sufficiently different context will produce different
+results:
+    >>> table = hl.utils.range_table(7, 1)
+    >>> table = table.filter(table.idx >= 2).annotate(x=hl.rand_unif(0, 1, seed=0))
+    >>> table.x.collect()
+    [0.20526631289173847,
+     0.6964416913998893,
+     0.6092952493383876,
+     0.6404026938964441,
+     0.5550464170615771]
+
+In fact, in this case we are getting the tail of
+    >>> table = hl.utils.range_table(7, 1).annotate(x=hl.rand_unif(0, 1, seed=0))
     >>> table.x.collect()
     [0.5820244750020055,
      0.33150686392731943,
      0.20526631289173847,
      0.6964416913998893,
-     0.6092952493383876]
+     0.6092952493383876,
+     0.6404026938964441,
+     0.5550464170615771]
+
+Reproducibility across sessions
+===============================
+
+The values of a random function are fully determined by three things:
+* The seed set on the function itself. If not specified, these are simply
+  generated sequentially.
+* Some data uniquely identifying the current position within a larger context,
+  e.g. Table, MatrixTable, or array. For instance, in a :func:`.range_table`,
+  this data is simply the row id, as suggested by the previous examples.
+* The global seed. This is fixed for the entire session, and can only be set
+  using the `global_seed` argument to :func:`.init`.
+
+To ensure reproducibility within a single hail session, it suffices to either
+manually set the seed on every random function call, or to call
+:func:`.reset_global_randomness` at the start of a pipeline, which resets the
+counter used to generate seeds.
+
+To ensure reproducibility across sessions, one must in addition specify the
+`global_seed` in :func:`.init`. If not specified, the global seed is chosen
+randomly.
+
 
 The seed can also be set globally using :func:`.set_global_seed`. This sets the
 seed globally for all subsequent Hail operations, and a pipeline will be
 guaranteed to have the same results if the global seed is set right beforehand:
 
-.. testsetup::
-    hl.reset_global_randomness()
-
-    >>> hl.set_global_seed(0)
-    >>> hl.eval(hl.array([hl.rand_unif(0, 1), hl.rand_unif(0, 1)]))  # doctest: +SKIP_OUTPUT_CHECK
-    [0.6830630912401323, 0.4035978197966855]
+    >>> hl.reset_global_randomness()
+    >>> hl.eval(hl.array([hl.rand_unif(0, 1), hl.rand_unif(0, 1)]))
+    [0.9828239225846387, 0.49094525115847415]
 
-    >>> hl.set_global_seed(0)
-    >>> hl.eval(hl.array([hl.rand_unif(0, 1), hl.rand_unif(0, 1)]))  # doctest: +SKIP_OUTPUT_CHECK
-    [0.6830630912401323, 0.4035978197966855]
+    >>> hl.reset_global_randomness()
+    >>> hl.eval(hl.array([hl.rand_unif(0, 1), hl.rand_unif(0, 1)]))
+    [0.9828239225846387, 0.49094525115847415]
 
 
 .. autosummary::