# Sheldon Ross 10: Exercise 3.22

Question: Suppose that independent trials, each of which is equally likely to have any of m outcomes, are performed until the same outcome occurs k consecutive times. If N denotes the number of trials, show that

E[N] = mk-1m-1

Simulation Solution for m = 10

Simulation Solution for m = 15

Code

```from sys import stdout
import sys
from mathematica.lists import EqualAll, Range
from mathematica.random_functions import RandomChoice

_folderPath = 'Your_Folder_Path_Here'
for _n in [10, 15]:
_steps = 10000
_subSteps = _steps // 100
__n = _n
_set = Range(_n)

for _k in range(2, 5):
_file = open(
_folderPath + 'sheldon_ross_10_exercise_3.22_' + str(_n) + '_' + str(
_k) + '.txt', 'w')

print("\n\nWriting data to the file :" + _file.name)

for i in range(0, _steps):
_sample = RandomChoice(_set, _k)
while True:
if EqualAll(_sample[-_k:]):
break
else:
_sample.append(RandomChoice(_set))
_file.write(_sample.__str__().replace("[", "{").replace("]", "}") + '\n')
if (i + 1) % _subSteps == 0:
stdout.write("\r" + ("\r" + ("#" * (i // _subSteps + 1)).ljust(100)) + " " + str(
i // _subSteps + 1) + " %")
stdout.flush()
print("\nClosing the file: " + _file.name)
_file.close()```

End of the post 😉

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