diff --git a/Lib/random.py b/Lib/random.py
--- a/Lib/random.py
+++ b/Lib/random.py
@@ -189,7 +189,10 @@
             if istart > 0:
                 if istart >= _maxwidth:
                     return self._randbelow(istart)
-                return _int(self.random() * istart)
+                result =  _int(self.random() * istart)
+                if result == istart:
+                    result = istart - 1
+                return result
             raise ValueError, "empty range for randrange()"
 
         # stop argument supplied.
@@ -213,7 +216,10 @@
 
             if width >= _maxwidth:
                 return _int(istart + self._randbelow(width))
-            return _int(istart + _int(self.random()*width))
+            offset =  _int(self.random() * width)
+            if offset == width:
+                offset = width - 1
+            return _int(istart + offset)
         if step == 1:
             raise ValueError, "empty range for randrange() (%d,%d, %d)" % (istart, istop, width)
 
@@ -233,7 +239,11 @@
 
         if n >= _maxwidth:
             return istart + istep*self._randbelow(n)
-        return istart + istep*_int(self.random() * n)
+
+        i = _int(self.random() * n)
+        if i == n:
+            i = n - 1
+        return istart + istep * i
 
     def randint(self, a, b):
         """Return random integer in range [a, b], including both end points.
@@ -272,7 +282,11 @@
 
     def choice(self, seq):
         """Choose a random element from a non-empty sequence."""
-        return seq[int(self.random() * len(seq))]  # raises IndexError if seq is empty
+        n = len(seq)
+        i = int(self.random() * n)
+        if i == n and n > 0:
+            i = n - 1
+        return seq[i]  # raises IndexError if seq is empty
 
     def shuffle(self, x, random=None):
         """x, random=random.random -> shuffle list x in place; return None.
@@ -288,7 +302,8 @@
         for i in reversed(xrange(1, len(x))):
             # pick an element in x[:i+1] with which to exchange x[i]
             j = _int(random() * (i+1))
-            x[i], x[j] = x[j], x[i]
+            if j < i:
+                x[i], x[j] = x[j], x[i]
 
     def sample(self, population, k):
         """Chooses k unique random elements from a population sequence.
@@ -332,17 +347,24 @@
             # mapping type so the other algorithm wouldn't work.
             pool = list(population)
             for i in xrange(k):         # invariant:  non-selected at [0,n-i)
-                j = _int(random() * (n-i))
+                limit = n - i
+                j = _int(random() * limit)
+                if j == limit:
+                    j = limit - 1
                 result[i] = pool[j]
-                pool[j] = pool[n-i-1]   # move non-selected item into vacancy
+                pool[j] = pool[limit-1] # move non-selected item into vacancy
         else:
             try:
                 selected = set()
                 selected_add = selected.add
                 for i in xrange(k):
                     j = _int(random() * n)
+                    if j == n:
+                        j = n - 1
                     while j in selected:
                         j = _int(random() * n)
+                        if j == n:
+                            j = n - 1
                     selected_add(j)
                     result[i] = population[j]
             except (TypeError, KeyError):   # handle (at least) sets