diff --git a/qiskit/primitives/base/base_estimator.py b/qiskit/primitives/base/base_estimator.py
index 1078091304ba..7406ff549b9a 100644
--- a/qiskit/primitives/base/base_estimator.py
+++ b/qiskit/primitives/base/base_estimator.py
@@ -22,7 +22,7 @@
 circuit and observable combinations.
 
 Following construction, an estimator is used by calling its :meth:`~.BaseEstimatorV2.run` method
-with a list of pubs (Primitive Unified Blocs). Each pub contains three values that, together,
+with a list of pubs (Primitive Unified Blocs). Each pub contains values that, together,
 define a computation unit of work for the estimator to complete:
 
 * a single :class:`~qiskit.circuit.QuantumCircuit`, possibly parametrized, whose final state we
@@ -30,9 +30,11 @@
 
 * one or more observables (specified as any :class:`~.ObservablesArrayLike`, including
   :class:`~.Pauli`, :class:`~.SparsePauliOp`, ``str``) that specify which expectation values to
-  estimate, denoted :math:`H_j`, and
+  estimate, denoted :math:`H_j`,
 
-* a collection parameter value sets to bind the circuit against, :math:`\theta_k`.
+* a collection parameter value sets to bind the circuit against, :math:`\theta_k`, and
+
+* optionally, a target precision `\epsilon` to meet or exceed for each estimate.
 
 Running an estimator returns a :class:`~qiskit.providers.JobV1` object, where calling
 the method :meth:`qiskit.providers.JobV1.result` results in expectation value estimates and metadata
@@ -77,8 +79,10 @@
     # calculate [ [<psi1(theta1)|H1|psi1(theta1)>,
     #              <psi1(theta3)|H3|psi1(theta3)>],
     #             [<psi2(theta2)|H2|psi2(theta2)>] ]
+    # aiming for precision to 0.02 on all estimates
     job2 = estimator.run(
-        [(psi1, [hamiltonian1, hamiltonian3], [theta1, theta3]), (psi2, hamiltonian2, theta2)]
+        [(psi1, [hamiltonian1, hamiltonian3], [theta1, theta3]), (psi2, hamiltonian2, theta2)],
+        precision=0.02
     )
     job_result = job2.result()
     print(f"The primitive-job finished with result {job_result}")
@@ -358,9 +362,8 @@ def run(self, pubs: Iterable[EstimatorPubLike], precision: float | None = None)
             pubs: An iterable of pub-like objects, such as tuples ``(circuit, observables)`` or
                   ``(circuit, observables, parameter_values)``.
             precision: The target precision for expectation value estimates of each
-                       run :class:`.EstimatorPub` that does not specify its own
-                       precision. If None the estimator's default precision value
-                       will be used.
+                       pub that does not specify its own precision. If ``None`` the 
+                       estimator's default precision value will be used.
 
         Returns:
             A job object that contains results.
diff --git a/qiskit/primitives/containers/estimator_pub.py b/qiskit/primitives/containers/estimator_pub.py
index 351f0f50564d..71df8bb11cbd 100644
--- a/qiskit/primitives/containers/estimator_pub.py
+++ b/qiskit/primitives/containers/estimator_pub.py
@@ -38,8 +38,6 @@ class EstimatorPub(ShapedMixin):
     estimator, if ``precision=None`` the estimator will determine the target precision.
     """
 
-    __slots__ = ("_circuit", "_observables", "_parameter_values", "_precision", "_shape")
-
     def __init__(
         self,
         circuit: QuantumCircuit,
@@ -55,6 +53,8 @@ def __init__(
             observables: An observables array.
             parameter_values: A bindings array, if the circuit is parametric.
             precision: An optional target precision for expectation value estimates.
+                This value takes precedence over any value owed by or supplied to an
+                estimator.
             validate: Whether to validate arguments during initialization.
         """
         super().__init__()
@@ -86,7 +86,10 @@ def parameter_values(self) -> BindingsArray:
 
     @property
     def precision(self) -> float | None:
-        """The target precision for expectation value estimates (optional)."""
+        """The target precision for expectation value estimates (optional).
+        
+        This value takes precedence over any value owed by or supplied to an estimator.
+        """
         return self._precision
 
     @classmethod
@@ -95,7 +98,7 @@ def coerce(cls, pub: EstimatorPubLike, precision: float | None = None) -> Estima
 
         Args:
             pub: A compatible object for coercion.
-            precision: an optional default precision to use if not
+            precision: An optional default precision to use if not
                        already specified by the pub-like object.
 
         Returns: