Three Prisoners’ Paradox
Mysteries regularly arise in probability because individuals have an inaccurate implication about a certain problem or on the grounds that, the definition is uncertain, prompting numerous interpretations. All the more explicitly, the intuitive perspective while solving a problem makes them appear as though a deficient rundown of the potential outcomes. Such probability paradox problems have always been the interest of scholars, statisticians, mathematicians, etc.
The three prisoner’s problem which is similar to the Monty hall problem was originally mentioned by Martin Gardner in his “Mathematical Games” column, 1959 edition of “Scientific American” and is another interesting counter-intuitive probability puzzle.
The Scenario
Natasha, Nathan, and Jeremy have been taken into custody for murder and are sentenced to death in three days. But the governor decides to pardon one of them as a part of the celebration of the national festival. The warden has been informed which one among the three is to be pardoned but is not allowed to tell. Natasha out of curiosity, begs the warden to let her know the identity of one of the others who are going to be executed. “If Nathan is to be pardoned, give me Jeremy’s name. If Jeremy is to be pardoned, give me Nathan’s name. And if I am to be pardoned, secretly flip a coin to decide whether to name Nathan or Jeremy”.
As the question was not directly about Natasha’s fate, the warden tells her that Nathan is to be executed. Natasha was pleased because she believed that her probability of surviving has gone up from 1/3 to 1/2, as it is now between her and Jeremy only. Natasha secretly tells Jeremy the news, who reasons that Natasha’s chance of being pardoned is unchanged at 1/3. In addition to that, he mentions that, rather his chance of surviving has doubled from 1/3 to 2/3. So, in reality, which prisoner is correct?
Initially, all the 3 prisoners have an equal chance of living, carrying a probability of 1/3. When it was announced that one of the three prisoners will be pardoned due to the special occasion in the state, Natasha asks the warden to know about the person who is going to be spared, as the warden knows this information. Natasha’s conditions while asking the warden are clearly represented in the tree diagram. “If Nathan is to be pardoned, give me Jeremy’s name. If Jeremy is to be pardoned, give me Nathan’s name.”. Branches of the tree diagram which explains situations if Nathan lives or Jeremy lives shows the processing of Natasha’s statement. After the announcement is made, Nathan and Jeremy have a chance of 1/3 to live. According to Natasha’s conditions, the other 2 prisoners have a direct trade-off, where if Nathan lives, then Jeremy dies; and when Jeremy lives, Nathan dies. But since Natasha also mentioned that if she is pardoned, the warden can flip a coin and tell one of the other 2 prisoner’s name. This is shown in the first branch of the tree diagram. The question to be analysed here is whether Natasha or Jeremy is right in their statements.
To justify the correct statement, consider the probabilities in the tree diagram considering cases where Nathan is executed. This scenario is taken into the calculation as it is the only direct, assured information obtained from the warden. With that, there are 2 cases where Natasha lives and Nathan dies, or Jeremy lives and Nathan dies. These 2 situations have probabilities of 1/6 and 1/3 respectively. This explains that after the information received that Nathan is going to be executed, Natasha has a probability of 1/6 to survive and Jeremy has a probability of 1/3 to survive. Natasha’s claim that she has improved her survival probability from 1/3 to 1/2 is falsified as her survival probability does not improve, but rather stays the same! On the other hand, Jeremy’s statement is right as he has twice the chance of survival that Natasha has ((1/6)*2 = 1/3 ). The three prisoner’s dilemma is interesting for probability learners due to this indirect reality behind it. Initially, every layman would vouch for Natasha and her statement. But after analysing the situation with ‘conditional’ or situational probabilities, Jeremy’s statement is justified and proven to be right.
Why is it Counter-Intuitive?
The most important reason for people to get it wrong is because we mostly focus on the situation from only Natasha’s perspective and not from the warden’s perspective who cannot reveal Natasha’s fate whatsoever that is.
As soon as the warden reveals about Nathan, Natasha jumps into the conclusion that now it’s just between her and Jeremy and believes that both of their chances of survival increased from 1/3 to 1/2. But an important thing to be noted here is that the warden may have tossed the coin if Natasha was to be released and that could be one way the warden gave Nathan’s name. Even if it was Jeremy, but the probabilities of the two events are not equal. But considering the situation that Nathan is to be executed and go forward with the information available, Jeremy has twice the chance that Natasha has to survive. It is ‘Counter-intuitive’ as it is contrary to the general expectations held by people like Natasha. Probabilities change according to conditions and there might be other ‘untold’ information which causes error in predicted results.
Solving using Bayes’ Theorem and Conditional Probability
In this scenario we have three events: A, B and C
Event A — Natasha will be pardoned
Event B — Nathan will be pardoned
Event C — Jeremy will be pardoned
Let ‘b’ be the event that the warden tells Natasha that Nathan is to be executed.
Therefore, we have:
P(A) = 1/3
P(B) = 1/3
P(C) = 1/3
Conditional Probability :
Probability of Nathan getting executed such that Natasha will be pardoned = P(b|A) = ½
Probability of Nathan getting executed such that Nathan will be pardoned = P(b|B) = 0
Probability of Nathan getting executed such that Jeremy will be pardoned = P(b|C) = 1
- Therefore, (1) using Bayes’ Theorem, the posterior probability of Natasha getting pardoned is,
Therefore, the probability of A being pardoned remains unchanged at 1/3
2. The Posterior Probability of Jeremy getting pardoned is:
Therefore, Jeremy’s chances of getting pardoned are doubled from 1/3 to 2/3
Conclusion
The main reason the probability of Natasha and Jeremy are unequal is because:
P(b|A) = 1/2 but
P(b|C) = 1
So, if Natasha is to be pardoned it is either because of:
1) Nathan being executed (P= ½)
2) Jeremy being executed (P= ½)
And hence, P(b|A) = ½
Whereas, if Jeremy is pardoned, the warden can only tell Natasha that Nathan is executed,
So, P(b|C) = 1
Therefore, in this way, we can conclude that Prisoner Jeremy was correct!