Three prisoners, A, B, and C, are locked in their cells. It is common knowledge that one of them is to be executed the next day and the others are to be pardoned. Only the Governor knows which one will be executed. Prisoner A asks the guard a favor: "Please ask the Governor who will be executed, and then take a message to one of my friends B or C to let him know that he will be pardoned in the morning." The Guard agrees, and comes back later and tells A that he gave the pardon message to B. What are A's chances of being executed, given this information? Show how you come to your answer mathematically. Failing that try running a little simulation
To determine the chances of Prisoner A being executed, we need to consider all possible scenarios and calculate the probabilities. Let's analyze the situation step by step.
First, let's assume that there are three possible scenarios, corresponding to each prisoner being the one to be executed. Without loss of generality, we can assume that A, B, and C represent these scenarios, respectively.
In scenario A: If Prisoner A is the one to be executed, then the Guard's response is truthful, and B will be pardoned tomorrow.
In scenario B: If Prisoner B is the one to be executed, then the Guard's response is a lie because B is not pardoned. So, the only possibility is that C will be pardoned.
In scenario C: If Prisoner C is the one to be executed, then again the Guard's response is a lie because C is not pardoned. Therefore, the only possibility is that B will be pardoned.
Now, let's analyze the probabilities associated with each scenario. Since the Governor is the only one who knows which prisoner will be executed, we have no prior information that allows us to assign probabilities to the scenarios. Therefore, we assume an equal probability of 1/3 for each scenario.
Given that the Guard informed Prisoner A that the message was given to B, we can now update the probabilities based on this information using Bayes' Theorem.
Let's define the following probabilities:
P(A) - Probability of scenario A (Prisoner A is executed)
P(B) - Probability of scenario B (Prisoner B is executed)
P(C) - Probability of scenario C (Prisoner C is executed)
Now, applying Bayes' Theorem:
P(A|B) - Probability of scenario A given the Guard's response is B
P(B|B) - Probability of scenario B given the Guard's response is B
P(C|B) - Probability of scenario C given the Guard's response is B
Bayes' Theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
P(B|B) = (P(B|B) * P(B)) / P(B)
P(C|B) = (P(B|C) * P(C)) / P(B)
Since P(B|B) = 0 (the Guard's response is a lie if Prisoner B is to be executed), we can simplify the equations:
P(A|B) = (P(B|A) * P(A)) / P(B)
P(C|B) = (P(B|C) * P(C)) / P(B)
Now, let's consider the probabilities as follows:
P(B|A) = 1 (the Guard always tells the truth when B is pardoned)
P(B|C) = 0 (the Guard always lies when C is to be executed)
P(A) = P(B) = P(C) = 1/3 (assuming equal probabilities for each scenario)
We need to find P(A|B), the probability that A is executed given the Guard's response is B:
P(A|B) = (1 * 1/3) / P(B)
P(A|B) = 1/3 / P(B)
To know P(B), the probability of the Guard's response being B, we need to consider the probability of the specific situation in which the Guard gives the pardon message to B.
From the initial three scenarios:
- In 1/3 of the possible situations, the Guard gives the message to A (scenario A).
- In 1/3 of the possible situations, the Guard gives the message to B (scenario B).
- In 1/3 of the possible situations, the Guard gives the message to C (scenario C).
So, only in the scenario B, the Guard gives the message to B. Therefore, P(B) = 1/3.
Now we can calculate P(A|B):
P(A|B) = 1/3 / (1/3)
P(A|B) = 1/3
Hence, given the information from the Guard's response that the message was given to B, Prisoner A's chances of being executed are 1/3.
To further confirm this, we can run a simulation:
Simulation:
Let's simulate the situation by programming a simple Python code:
```python
import random
def simulate_execution():
prisoners = ['A', 'B', 'C']
executed_prisoner = random.choice(prisoners)
message_recipient = random.choice(['B', 'C'])
guard_response = 'B'
return executed_prisoner == 'A'
num_simulations = 1000000
executed_count = 0
for _ in range(num_simulations):
if simulate_execution():
executed_count += 1
execution_probability = executed_count / num_simulations
print(f"Simulation result: The probability of A being executed is approximately {execution_probability}")
```
By running the simulation for a large number of iterations (e.g., 1,000,000), the result will approximate the theoretical probability we derived mathematically.