How do I set an element of a doubly nested list in Python?

Question

I run the following lines in the Python interpreter and I get:

>>> s = [[0]*5]*5
>>> s
[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]
>>> s[2][4] = 1
>>> s
[[0, 0, 0, 0, 1], [0, 0, 0, 0, 1], [0, 0, 0, 0, 1], [0, 0, 0, 0, 1], [0, 0, 0, 0, 1]]

However I want the last line to be:

[[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 1], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]

Why is this not happening?