diff --git a/CHANGES b/CHANGES
index f57e5ca13..6d6810c06 100644
--- a/CHANGES
+++ b/CHANGES
@@ -8,6 +8,8 @@
 
   * Fix win32_spawn leaking dev_null fd in the parent process. (#906, Antonin Décimo)
   * Prefer SetHandleInformation to DuplicateHandle in set_close_on_exec for sockets. DuplicateHandle mustn't be used on sockets. (#907, Antonin Décimo)
+  * Lwt.pick and Lwt.choose select preferentially failed promises as per
+  documentation (#856, #874, Raman Varabets)
 
 ===== 5.5.0 =====
 
diff --git a/src/core/lwt.ml b/src/core/lwt.ml
index 9dd731bf7..ec5858246 100644
--- a/src/core/lwt.ml
+++ b/src/core/lwt.ml
@@ -2642,15 +2642,28 @@ struct
      [choose]/[pick] implementation, which may actually be optimal anyway with
      Flambda. *)
 
-  let count_resolved_promises_in (ps : _ t list) =
-    let accumulate total p =
-      let Internal p = to_internal_promise p in
-      match (underlying p).state with
-      | Fulfilled _ -> total + 1
-      | Rejected _ -> total + 1
-      | Pending _ -> total
+  let count_resolved_promises_in (ps : 'a t list) =
+    let rec count_and_gather_rejected total rejected ps =
+       match ps with
+       | [] -> Result.Error (total, rejected)
+       | p :: ps ->
+            let Internal q = to_internal_promise p in
+            match (underlying q).state with
+            | Fulfilled _ -> count_and_gather_rejected total rejected ps
+            | Rejected _ -> count_and_gather_rejected (total + 1) (p :: rejected) ps
+            | Pending _ -> count_and_gather_rejected total rejected ps
+    in
+    let rec count_fulfilled total ps =
+       match ps with
+       | [] -> Result.Ok total
+       | p :: ps ->
+            let Internal q = to_internal_promise p in
+            match (underlying q).state with
+            | Fulfilled _ -> count_fulfilled (total + 1) ps
+            | Rejected _ -> count_and_gather_rejected 1 [p] ps
+            | Pending _ -> count_fulfilled total ps
     in
-    List.fold_left accumulate 0 ps
+    count_fulfilled 0 ps
 
   (* Evaluates to the [n]th promise in [ps], among only those promises in [ps]
      that are resolved. The caller is expected to ensure that there are at
@@ -2704,7 +2717,7 @@ struct
       invalid_arg
         "Lwt.choose [] would return a promise that is pending forever";
     match count_resolved_promises_in ps with
-    | 0 ->
+    | Result.Ok 0 ->
       let p = new_pending ~how_to_cancel:(propagate_cancel_to_several ps) in
 
       let callback result =
@@ -2718,17 +2731,20 @@ struct
 
       to_public_promise p
 
-    | 1 ->
+    | Result.Ok 1 ->
       nth_resolved ps 0
 
-    | n ->
+    | Result.Ok n ->
+      nth_resolved ps (Random.State.int (Lazy.force prng) n)
+
+    | Result.Error (n, ps) ->
       nth_resolved ps (Random.State.int (Lazy.force prng) n)
 
   let pick ps =
     if ps = [] then
       invalid_arg "Lwt.pick [] would return a promise that is pending forever";
     match count_resolved_promises_in ps with
-    | 0 ->
+    | Ok 0 ->
       let p = new_pending ~how_to_cancel:(propagate_cancel_to_several ps) in
 
       let callback result =
@@ -2743,13 +2759,17 @@ struct
 
       to_public_promise p
 
-    | 1 ->
+    | Ok 1 ->
       nth_resolved_and_cancel_pending ps 0
 
-    | n ->
+    | Ok n ->
       nth_resolved_and_cancel_pending ps
         (Random.State.int (Lazy.force prng) n)
 
+    | Error (n, qs) ->
+      List.iter cancel ps;
+      nth_resolved qs (Random.State.int (Lazy.force prng) n)
+
 
 
   (* If [nchoose ps] or [npick ps] found all promises in [ps] pending, the
diff --git a/test/core/test_lwt.ml b/test/core/test_lwt.ml
index 7c14f5ecd..803b02364 100644
--- a/test/core/test_lwt.ml
+++ b/test/core/test_lwt.ml
@@ -2274,11 +2274,10 @@ let choose_tests = suite "choose" [
   end;
 
   test "multiple resolved" begin fun () ->
-    (* This is run in a loop to exercise the internal PRNG. *)
-    let outcomes = Array.make 3 0 in
+    (* This is run in a loop to check that it consistently returns the failed
+       result as per documentation. *)
     let rec repeat n =
-      if n <= 0 then ()
-      else
+      n <= 0 || begin
         let p =
           Lwt.choose
             [fst (Lwt.wait ());
@@ -2286,19 +2285,15 @@ let choose_tests = suite "choose" [
              Lwt.fail Exception;
              Lwt.return "bar"]
         in
-        begin match Lwt.state p with
-        | Lwt.Return "foo" -> outcomes.(0) <- outcomes.(0) + 1
-        | Lwt.Fail Exception -> outcomes.(1) <- outcomes.(1) + 1
-        | Lwt.Return "bar" -> outcomes.(2) <- outcomes.(2) + 1
+        match Lwt.state p with
+        | Lwt.Return "foo" -> false
+        | Lwt.Fail Exception -> repeat (n - 1)
+        | Lwt.Return "bar" -> false
         | _ -> assert false
         end [@ocaml.warning "-4"];
-        repeat (n - 1)
     in
-    let count = 1000 in
-    repeat count;
-    Lwt.return
-      (outcomes.(0) > 0 && outcomes.(1) > 0 && outcomes.(2) > 0 &&
-       outcomes.(0) + outcomes.(1) + outcomes.(2) = count)
+    let count = 100 in
+    Lwt.return (repeat count)
   end;
 
   test "pending" begin fun () ->
@@ -2982,31 +2977,29 @@ let pick_tests = suite "pick" [
   end;
 
   test "multiple resolved" begin fun () ->
-    (* This is run in a loop to exercise the internal PRNG. *)
-    let outcomes = Array.make 3 0 in
+    (* This is run in a loop to check that it consistently returns the failed
+       result as per documentation. *)
     let rec repeat n =
-      if n <= 0 then ()
-      else
+      n <= 0 || begin
+        let (waiter, _) = Lwt.task () in
         let p =
           Lwt.pick
-            [fst (Lwt.wait ());
+            [waiter;
              Lwt.return "foo";
              Lwt.fail Exception;
              Lwt.return "bar"]
         in
-        begin match Lwt.state p with
-        | Lwt.Return "foo" -> outcomes.(0) <- outcomes.(0) + 1
-        | Lwt.Fail Exception -> outcomes.(1) <- outcomes.(1) + 1
-        | Lwt.Return "bar" -> outcomes.(2) <- outcomes.(2) + 1
+        match Lwt.state p with
+        | Lwt.Return "foo" -> false
+        | Lwt.Fail Exception ->
+            Lwt.state waiter = Lwt.Fail Lwt.Canceled
+            && repeat (n - 1)
+        | Lwt.Return "bar" -> false
         | _ -> assert false
         end [@ocaml.warning "-4"];
-        repeat (n - 1)
     in
-    let count = 1000 in
-    repeat count;
-    Lwt.return
-      (outcomes.(0) > 0 && outcomes.(1) > 0 && outcomes.(2) > 0 &&
-       outcomes.(0) + outcomes.(1) + outcomes.(2) = count)
+    let count = 100 in
+    Lwt.return (repeat count)
   end;
 
   test "pending" begin fun () ->