diff --git a/Lib/random.py b/Lib/random.py
index 38c4a54052..8322144e8d 100644
--- a/Lib/random.py
+++ b/Lib/random.py
@@ -484,13 +484,11 @@ def choices(self, population, weights=None, *, cum_weights=None, k=1):
         the selections are made with equal probability.
 
         """
-        random = self.random
         n = len(population)
         if cum_weights is None:
             if weights is None:
-                floor = _floor
-                n += 0.0    # convert to float for a small speed improvement
-                return [population[floor(random() * n)] for i in _repeat(None, k)]
+                randbelow = self._randbelow
+                return [population[randbelow(n)] for i in _repeat(None, k)]
             try:
                 cum_weights = list(_accumulate(weights))
             except TypeError:
@@ -504,13 +502,19 @@ def choices(self, population, weights=None, *, cum_weights=None, k=1):
             raise TypeError('Cannot specify both weights and cumulative weights')
         if len(cum_weights) != n:
             raise ValueError('The number of weights does not match the population')
-        total = cum_weights[-1] + 0.0   # convert to float
+        total = cum_weights[-1]
         if total <= 0.0:
             raise ValueError('Total of weights must be greater than zero')
         if not _isfinite(total):
             raise ValueError('Total of weights must be finite')
         bisect = _bisect
         hi = n - 1
+        if isinstance(total, int):
+            randbelow = self._randbelow
+            return [population[bisect(cum_weights, randbelow(total), 0, hi)]
+                    for i in _repeat(None, k)]
+        total += 0.0   # convert to float
+        random = self.random
         return [population[bisect(cum_weights, random() * total, 0, hi)]
                 for i in _repeat(None, k)]
 
diff --git a/Lib/test/test_random.py b/Lib/test/test_random.py
index c2dd50b981..3f24e9a7ed 100644
--- a/Lib/test/test_random.py
+++ b/Lib/test/test_random.py
@@ -303,6 +303,17 @@ def test_choices_subnormal(self):
         choices = self.gen.choices
         choices(population=[1, 2], weights=[1e-323, 1e-323], k=5000)
 
+    def test_choices_no_loss_of_precision(self):
+        # Make sure big odd numbers are included
+        K = 100
+        for n in range(1, 64):
+            try:
+                choices = random.choices(range(2**n), k=K)
+            except OverflowError:
+                continue
+            parities = [x % 2 for x in choices]
+            self.assertNotEqual(parities, [0]*K, msg=f"{n} bit choices")
+
     def test_choices_with_all_zero_weights(self):
         # See issue #38881
         with self.assertRaises(ValueError):