**Python Puzzles**

You are locked in a jail cell and demand to be released because you are innocent. The warden tells you he will release you if you can win 2 straight games of checkers. But you must play at least one game against the warden and one game against the sheriff. You know the sheriff plays a better game of checkers then the warden.

Do you have a better probability of winning if you play [Warden -> Sheriff -> Warden] or [Sheriff -> Warden -> Sheriff]? Remember, the sheriff is a better checkers player then the warden, and you must win 2 straight games.

You have a higher probability of winning 2 straight games playing [Sheriff -> Warden -> Sheriff].

The probability changes depends on the probability variables you set for the warden and the sheriff. But given you must win the second game in order to get out of jail, it is best to play the weaker opponent second and hope you can win 1 of 2 from the tougher opponent.

import numpy as np # Sheriff's and warden's win probabilities sheriff = .75 warden = .5 # List to hold the number of scenarios where Sheriff wins in each game scenario1 = [] # List to hold the number of scenarios where warden wins in each game scenario2 = [] # Loop through 1 million games for _ in range(1_000_000): # Game scenario: sheriff -> warden -> sheriff winning_count = 0 # Play 1 sheriff_play1 = np.random.binomial(1, sheriff, 1).item() winning_count += 1 if sheriff_play1 == 0 else winning_count # Play 2 warden_play1 = np.random.binomial(1, warden, 1).item() winning_count = winning_count + 1 if warden_play1 == 0 else 0 # If Sheriff wins 2 out of 3 plays, add 1 to scenario1 if winning_count == 2: scenario1.append(1) # Play 3 sheriff_play2 = np.random.binomial(1, sheriff, 1).item() winning_count = winning_count + 1 if sheriff_play2 == 0 else 0 # If Sheriff wins 2 out of 3 plays, add 1 to scenario1 if winning_count >= 2: scenario1.append(1) # Game scenario: sheriff -> warden -> sheriff winning_count = 0 # Play 1 warden_play1 = np.random.binomial(1, warden, 1).item() winning_count += 1 if warden_play1 == 0 else winning_count # Play 2 sheriff_play1 = np.random.binomial(1, sheriff, 1).item() winning_count = winning_count + 1 if sheriff_play1 == 0 else 0 # If Sheriff wins 2 out of 3 plays, add 1 to scenario2 if winning_count == 2: scenario2.append(1) winning_count = 0 # Play 3 warden_play2 = np.random.binomial(1, warden, 1).item() winning_count = winning_count + 1 if warden_play2 == 0 else 0 # If Sheriff wins 2 out of 3 plays, add 1 to scenario2 if winning_count >= 2: scenario2.append(1) # Print the number of scenarios where Sheriff wins in each game print("Number of scenarios where Sheriff wins (scenario1):", len(scenario1)) print("Number of scenarios where Sheriff wins (scenario2):", len(scenario2))