diff --git a/Lib/random.py b/Lib/random.py
index 1e0dcc87ed..e3dd48aa11 100644
--- a/Lib/random.py
+++ b/Lib/random.py
@@ -300,9 +300,16 @@ class Random(_random.Random):
                 x[i], x[j] = x[j], x[i]
         else:
             _int = int
+            _min = min
             for i in reversed(range(1, len(x))):
-                # pick an element in x[:i+1] with which to exchange x[i]
-                j = _int(random() * (i+1))
+                # Pick an element in x[:i+1] with which to exchange x[i].
+                #
+                # The min() clamp was added for Issue #24567 to handle
+                # a rare and platform dependent situation where j could
+                # rounded to an invalid index.  Periodically, we should
+                # assess whether this clamp is still necessary since it
+                # obscures and slows down the code.
+                j = _min(i, _int(random() * (i+1)))
                 x[i], x[j] = x[j], x[i]
 
     def sample(self, population, k):
@@ -373,9 +380,16 @@ class Random(_random.Random):
         random = self.random
         if cum_weights is None:
             if weights is None:
+                # The min() clamp was added for Issue #24567 to handle
+                # a rare and platform dependent situation where j could
+                # rounded to an invalid index.  Periodically, we should
+                # assess whether this clamp is still necessary since it
+                # obscures and slows down the code.
                 _int = int
+                _min = min
                 total = len(population)
-                return [population[_int(random() * total)] for i in range(k)]
+                return [population[_min(_int(random() * total), total)]
+                        for i in range(k)]
             cum_weights = list(_itertools.accumulate(weights))
         elif weights is not None:
             raise TypeError('Cannot specify both weights and cumulative weights')
@@ -383,7 +397,14 @@ class Random(_random.Random):
             raise ValueError('The number of weights does not match the population')
         bisect = _bisect.bisect
         total = cum_weights[-1]
-        return [population[bisect(cum_weights, random() * total)] for i in range(k)]
+        hi = len(population) - 1
+        # Clamping the bisect upper limit to *hi* was added for Issue
+        # #24567 to handle a rare and platform dependent situation where
+        # random()*total could slightly exceed the value of total.
+        # Periodically, we should assess whether this clamp is still
+        # necessary since it obscures and slows down the code.
+        return [population[bisect(cum_weights, random() * total, 0, hi)]
+                for i in range(k)]
 
 ## -------------------- real-valued distributions  -------------------