Python randomly generated walks give same outcome when graphed

from pylab import *

no_steps = 10000

number = random()
position = zeros(no_steps)
position[0] = 0
time = zeros(no_steps)
time[0] = 0

for i in range(1, no_steps):
    time[i] = time[i-1] + 1
    if number >= 0.5:
        position[i] = position[i-1] + 1
        number = random()
    else:
        position[i] = position[i-1] - 1
        number = random()
plot(time, position)

number2 = random()
position2 = zeros(no_steps)
position2[0] = 0
time2 = zeros(no_steps)
time2[0] = 0
for t2 in range(1, no_steps):
    time2[t2] = time[t2-1] + 1
    if number2 >= 0.5:
        position2[t2] = position2[t2-1] + 1
        number2 = random()
    else:
        position2[t2] = position[t2-1] - 1
        number2 = random()
plot(time2,position2)

This is supposed to generate random walks by generating a random number each time and checking the conditions. Therefore I assumed that if it works for one walk I can just add more of the same and put them all on the same graph at the end. However, apparently that's not how this works and the graphs that do end up being plotted are extremely similar with the difference in the positions being one of -2 for some reason. The code if I run the blocks separately from their own program will generate two completely different walks, it's just when I put them together that it stops working as intended. What exactly am I missing?