diff --git a/pkgs/data-pkgs/data-doc/data/scribblings/enumerate.scrbl b/pkgs/data-pkgs/data-doc/data/scribblings/enumerate.scrbl
index edc603a6f6..000eb195ee 100644
--- a/pkgs/data-pkgs/data-doc/data/scribblings/enumerate.scrbl
+++ b/pkgs/data-pkgs/data-doc/data/scribblings/enumerate.scrbl
@@ -672,6 +672,19 @@ An @tech{enumeration} of S-expressions.
 This library defines some library @tech{enumerations} built on
 @racketmodname[data/enumerate].
 
+@defproc[(random-index [e enum?])
+         exact-nonnegative-integer?]{
+
+Returns a random index into @racket[e]. This works regardless of the
+size of @racket[e], unlike @racket[(random (size e))].
+
+@examples[#:eval the-eval
+(random (size nat/e))
+(random (size (below/e 5000000000)))
+(random-index nat/e)
+(random-index (below/e 5000000000))
+]}
+
 @defproc[(permutations-of-n/e [n exact-nonnegative-integer?])
          enum?]{
 
diff --git a/pkgs/data-pkgs/data-enumerate-lib/data/enumerate.rkt b/pkgs/data-pkgs/data-enumerate-lib/data/enumerate.rkt
index 13e3a69f9b..9f0fb432a7 100644
--- a/pkgs/data-pkgs/data-enumerate-lib/data/enumerate.rkt
+++ b/pkgs/data-pkgs/data-enumerate-lib/data/enumerate.rkt
@@ -106,7 +106,7 @@
    (-> enum? (-> any/c enum?)
        enum?)]
   [dep2/e
-   (-> nat? enum? (-> any/c enum?)
+   (-> extended-nat/c enum? (-> any/c enum?)
        enum?)]
   [fold-enum
    (-> (-> list? any/c enum?)
diff --git a/pkgs/data-pkgs/data-enumerate-lib/data/enumerate/lib.rkt b/pkgs/data-pkgs/data-enumerate-lib/data/enumerate/lib.rkt
index cd8b8137e8..492754b4d4 100644
--- a/pkgs/data-pkgs/data-enumerate-lib/data/enumerate/lib.rkt
+++ b/pkgs/data-pkgs/data-enumerate-lib/data/enumerate/lib.rkt
@@ -1,9 +1,30 @@
 #lang racket/base
 (require racket/contract/base
          data/enumerate
+         math/base         
+         math/distributions
          math/number-theory
          racket/generator)
 
+;; pick-an-index : ([0,1] -> Nat) ∩ (-> Nat)
+(define (random-natural-w/o-limit [prob-of-zero 0.01])
+  (max (random-natural/no-mean prob-of-zero)
+       (random-natural/no-mean prob-of-zero)
+       (random-natural/no-mean prob-of-zero)))
+
+;; random-natural/no-mean : [0,1] -> Nat
+(define (random-natural/no-mean prob-of-zero)
+  (define x (sample (geometric-dist prob-of-zero)))
+  (define m1 (expt 2 (exact-floor x)))
+  (define m0 (quotient m1 2))
+  (random-integer m0 m1))
+
+(define (random-index e)
+  (define k (size e))
+  (if (infinite? k)
+      (random-natural-w/o-limit)
+      (random-natural k)))
+
 (define (BPP-digits N)
   (let loop ([8Pi -8])
     (define 8i
@@ -168,6 +189,8 @@
 
 (provide
  (contract-out
+  [random-index
+   (-> enum? exact-nonnegative-integer?)]
   [infinite-sequence/e
    (-> enum? enum?)]
   [permutations/e