[Python] Why does my code give not the expected value at the prisoners problem?

I have a question about the "Prisoners Problem" and my tackeling of that. I made a little Project simulating it in Python, and the results I'm getting are not what I was expecting. I'm getting around a 45-55% succsessrate, while the blank math gets a round about 33% succsessrate.

Here is my Code for that:

def GenerateBoxes():
import random
from random import sample

# Generate a list of 100 unique random numbers between 1 and 100
random_values = sample(range(1, 101), 100)

# Create a dictionary (boxes) with 100 different keys and the generated random values
boxes = {i: value for i, value in enumerate(random_values, start=1)}

return boxes




def SimulateExperiment(timesGiven):

# a list of wether the 100 prisoners escape or not, represented as a bool
sucsesses = []

# how often the experiment should be repeated
times = timesGiven
x = 0
while x < times:
boxes = GenerateBoxes()
escape = True;

p_num = 1
temp_p_num = 1
for p_num in range(1, 100):
k = 1
while k < 100:
if k <= 50 and temp_p_num == boxes[p_num]:
break
elif k > 50 and temp_p_num == boxes[p_num]:
escape = False
break
else:
temp_p_num = boxes[temp_p_num]
k += 1

p_num += 1

if escape == True:
sucsesses.append(x)

x += 1

return sucsesses

times = 100000

sucsessrate = SimulateExperiment(times)
sucsessPercent = (len(sucsessrate) / times) * 100
print( "Sucsessrate: " + len(sucsessrate).__str__() + " out of " + times.__str__() + ": " + sucsessPercent.__str__() + "%")


Do I have a thinking error or why do I get different results?

Thank you for reading!

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