diff --git a/doc/releases/changelog-dev.md b/doc/releases/changelog-dev.md
index 00011f80627..a48d37fd7ab 100644
--- a/doc/releases/changelog-dev.md
+++ b/doc/releases/changelog-dev.md
@@ -294,6 +294,9 @@
 
 <h3>Bug fixes 🐛</h3>
 
+* Avoid bounded value failures due to numerical noise with calls to `np.random.binomial`.
+  [(#5447)](https://github.com/PennyLaneAI/pennylane/pull/5447)
+
 * Using `@` with legacy Hamiltonian instances now properly de-queues the previously existing operations.
   [(#5454)](https://github.com/PennyLaneAI/pennylane/pull/5455)
 
diff --git a/pennylane/devices/qubit/apply_operation.py b/pennylane/devices/qubit/apply_operation.py
index ca9db1063ad..61ea0956a47 100644
--- a/pennylane/devices/qubit/apply_operation.py
+++ b/pennylane/devices/qubit/apply_operation.py
@@ -265,7 +265,15 @@ def apply_mid_measure(
         return np.zeros_like(state)
     wire = op.wires
     probs = qml.devices.qubit.measure(qml.probs(wire), state)
-    sample = np.random.binomial(1, probs[1])
+
+    try:  # pragma: no cover
+        sample = np.random.binomial(1, probs[1])
+    except ValueError as e:  # pragma: no cover
+        if probs[1] > 1:  # MachEps error, safe to catch
+            sample = np.random.binomial(1, np.round(probs[1], 15))
+        else:  # Other general error, continue to fail
+            raise e
+
     mid_measurements[op] = sample
     if op.postselect is not None and sample != op.postselect:
         return np.zeros_like(state)