diff --git a/axelrod/strategies/rand.py b/axelrod/strategies/rand.py
index 0a7593039..077b5a048 100644
--- a/axelrod/strategies/rand.py
+++ b/axelrod/strategies/rand.py
@@ -36,6 +36,8 @@ def __init__(self, p=0.5):
         """
         Player.__init__(self)
         self.p = p
+        if p in [0, 1]:
+            self.classifier['stochastic'] = False
 
     def strategy(self, opponent):
         return random_choice(self.p)
diff --git a/axelrod/tests/unit/test_rand.py b/axelrod/tests/unit/test_rand.py
index 38983bd3e..3b7829f44 100644
--- a/axelrod/tests/unit/test_rand.py
+++ b/axelrod/tests/unit/test_rand.py
@@ -28,3 +28,9 @@ def test_strategy(self):
         self.first_play_test(C, random_seed=1)
         self.first_play_test(D, random_seed=2)
         self.responses_test(response_1, response_2, [C], random_seed=1)
+
+    def test_deterministic_classification(self):
+        """Test classification when p is 0 or 1"""
+        for p in [0, 1]:
+            player = axelrod.Random(p)
+            self.assertFalse(player.classifier['stochastic'])