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## The prisoner

Posted by Chris on December 5, 2010 – 8:11 am

As I’m a glutton for punishment, here’s another classic.

Out of three prisoners (whose names are Alan, Bill and Charles) scheduled to be put to death, it has been decided that one of them will be pardoned.  The pardoned prisoner is chosen randomly.

Alan asks the warden to tell him the name of one of the others who will be executed.  The warden is under strict instructions to not reveal the fate of a prisoner to that prisoner.  The warden (who knows who is going to be pardoned and speaks truthfully) reveals to Alan that Bill is going to be executed.

What is the probability that Charles is going to be executed?

What is the probabilty that Alan is going to be executed?

Hint: neither probability is 1/2.

Much later (July 2011): The question can only sensibly be answered if you assume that the warden is unbiased. i.e. If Alan is going to be pardoned, it is equally likely that the warden will name Bill or Charles.

This post is under “Tom” and has 39 respond so far.
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### 39 Responds so far- Add one»

1. 1. slavy Said：

Chris, I smell more than 60 (incorrect) replies to this post and you will be in a much bigger hell convincing the people in the answer than the B/W card problem. Good luck

2. 2. Chris Said：

Hi slavy. LOL, more like 15. Please confirm that I haven’t made an error due to the way I’ve phrased the question.

3. 3. Peter Said：

I think that both Alan and Charles odds are 2/3. Unlikely to be right as it sounds too easy.

4. 4. bree Said：

Since they are all schedualed to be put to death, wouldn’t they all have a 100% chance of dying?

5. 5. Jim Said：

Allen and Charles both have a 25% chance.

6. 6. Kate Said：

Well, they’re both going to die eventually, and the riddle doesn’t ask the probability of them being executed, just the probability of them dying. So, they both have a 100% chance of dying.

7. 7. slavy Said：

Hi Chris. There is always an option that I myself am wrong, too, but the problem looks fine to me and is equivalent to the other more popular version (the one with winning something from a show).

8. 8. Chris Said：

Thanks slavy. At least they’re not getting it wrong in the usual boring old way

9. 9. Karl Sharman Said：

bree is correct. All of the prisoners will die. 100% probability. 2 of them will be executed, and one will probably die of old age, unless some unforeseen accident befalls him before his appointed time.
If you want specifics….. the tiger is behind door number 3.

10. 10. Shelley Said：

Is this a trick question? Cause their chances of dying are 100%, though there chances of being executed are lower. Are the chances of being executed 2 in 3?

11. 11. Shelley Said：

Wait, no I take the 2 in 3 back. Let me crunch some numbers and get back to you.

12. 12. Chris Said：

I’ve just changed “die” to “be executed” – the original wording was a slip.

13. 13. Chris Said：

Shelley, I hope you’re not going to descend to using a computer!

14. 14. Shelley Said：

I seriously don’t know how I would use a computer to solve this, you mean just google it?

Anyway, I think Allan has a 2 in 3 chance, but Charles only has a 1 in 3 chance. I won’t say how I got it yet; I don’t want to spoil someone’s fun if I’m right. This was a really challenging problem, it gave me a major headache.

15. 15. Chris Said：

Hi Shelley. You said “crunch some numbers”, so I assumed you meant simulating – an act for you you would have received almost no credit.

I’m surprised that you struggled, especially afer the coin flipping one.

Anyway, you’re the first with both parts right. Thank you for holding back your method. But you may publish it whenever you like.

16. 16. Shelley Said：

Yay I don’t know why I had more trouble with this one, and I’ll post how I did it sometime tomorrow.

17. 17. andrew Said：

well i would say that charles would have 33% chance of being executed. and as for Alan i think i could safely say that he also has a 33% of being executed

18. 18. bree Said：

So am i wrong or right? no one answered me?

19. 19. Random Guy Said：

@bree
I dont think youre correct, since 1 of them will be pardoned.

20. 20. mike Said：

i think the problem is slightly off? the variation or “classic” way is the game show and the 3 doors.

But answer in that riddle is that as the game show host takes a door away, it does not change the odds of the choice of door the contestant made.

In this riddle, what was the 1st choice the contestant made?

I agree w/ Shelly’s answer but is the real answer Alan has 1/3rd and Chris has 2/3rds, or vice versa?

21. 21. Jaras Said：

This is similar to donkey and ferrari game show.
to Alan prob still 2/3 while chales. Prob reduced to 1/3

22. 22. Chris Said：

With the original wording of the problem, those of you who said 100% were right. But that was an unintentional trick question.

For the question as it now reads (edited at post 12), Alan has a 2/3 probability of being executed throughout, but Charles’ probability of being executed shifts from 2/3 to 1/3 when we realise that Bill has had it. (The exact point at which the probability changes, or even if it did, is probably open to philosophical debate).

23. 23. Chris Said：

It seems that both slavy and I were over-pessimistic about the replies that this problem would elicit.

Perhaps my re-wording of the problem (chance of execution rather that survival and skipping the option of being able to swap with Charles) had something to do with that.

24. 24. Kenny Said：

“Alan asks the warden to tell him the name of one of the others who will be executed” So obviously Alan is one of the person being killed and the warden said the other is Bill so it leaves charles at 0% chance of being executed..

25. 25. Chris Said：

Hi Kenny. You’ve not read the question properly.

26. 26. Cory Said：

y would the warden tell the prisoner who is getting killed??

Bill is getting the pardon:P

27. 27. Cory Said：

Think out side of the box

28. 28. Chris Said：

Think outside the box by all means. But don’t change its contents.

29. 29. Kenny Said：

What do you mean i didnt read the question properly? Its true though he asked the warden for the name of the OTHER person being excuted therefore he is stating that he is one of the person being executed.

30. 30. Chris Said：

Hi Kenny. My bad – I should have said “You haven’t comprehended the question correctly”.

The question states that “the warden is under strict instructions to not reveal the fate of a prisoner to that prisoner” – i.e. The warden can’t tell Alan his fate regardless of whether it’s good or bad. You somehow take this to mean that Alan is definitely going to be executed.

Your post (29) establishes this comprehension failure once again. You have changed “one of the others” to “the other”. The question, as written, provides no clue about Alan’s fate, whereas your re-write guarantees Alan’s early demise.

31. 31. Chris Said：

Here’s my analysis:

I’ll abbreviate the names to A, B, C and W. The three possible scenarios are:
1. A is going to be pardoned, then W could say B or C.
2. B is going to be pardoned, then W has to say C.
3. C is going to be pardoned, then W has to say B.

In 1, A is to be pardoned and the unnamed man is to be executed.
In 2, A is to be executed and the unnamed man is to be pardoned.
In 3, A is to be executed and the unnamed man is to be pardoned.

Each scenario is equally likely, with probability 1/3. So there is a 2/3
probabilty that Alan is to be executed and a 1/3 probability that the
unnamed man is to be executed. In this problem the unnamed man
is Charles.

Note that Alan’s chances haven’t changed one iota by the warden
naming Bill. But Alan’s and our knowledge of the probability of Charles’
fate has improved from 2/3 to 1/3 of Charles being executed.

I like this analysis, partly because it is entirely my own, but mainly
because it avoids all question of any bias that the warden may have
when selecting which name to provide in scenario 1. Some versions of the
problem say that the warden might toss a coin to decide. The fact is that
it doesn’t matter how he decides.

Many versions of the problem allow you the chance to swap places with the unnamed man. I’d definitely take advantage of that offer.

This problem is equivalent to the Monty Hall (game show) problem that slavy hinted at.

32. 32. Chris Said：

No challengers! OK, it looks like I’ll have to challenge myself as I’m suffering from cognitive dissonance.

My doubt arisies from the following consideration. If the warden has a bias to always name Charles in scenario 1, then I’d normally have said that if the warden actually says Bill, that we must be in scenario 3 and so Alan is definitely for the chop and Charles is definitely going to be pardoned.

This sure seems like a paradox to me.

If I re-introduce the usual extra that in scenario 1, the warden flips a coin to decide which name to give to Alan, then we get (when he says Bill – so we can’t be in scenario 2) then the (conditional) probability of being in scenario 3 is 2/3, and of being in scenario 1 is 1/3. In which case we have the “usual” result that Alan has a 2/3 chance of being executed, and Charles only has a 1/3 chance of being executed.

I conclude that my original luvverly logical argument has a serious flaw, and that any bias the warden has of naming a prisoner in scenario 1 is relevant.

It’s no wonder that the statisticians have major philosophical camps about the interpretation of probablity matters.

33. 33. Chris Said：

I know what the fault is: “unnamed man” is not [a] well defined [set]. It’s a bit like the Barber in the Barber’s Paradox (proper version). It’s been over a year since I worked out the first logic and I’ve only just realised the flaw.

For a little more completeness: If the warden had a bias to always say Bill in scenario 1, then when he actually says it, we must be in scenario 1 or 3 with equal likelihood. Then Alan’s chance of execution is 1/2 as is Charles’s. This is the only case where swapping wouldn’t give you an advantage – but you’d be no worse of.

34. 34. Ipwnznoobz Said：

Alan is going to live so 0/1 and charles is going to die 1/1

read the words carefully, especially around who the warden can and can’t tell

35. 35. Chris Said：

Hi Ipwnznoobz. I can’t see how you worked that out.

The Warden cannot tell Alan his fate, but he can tell Alan either Bill’s or Charles’s fate (but not both as that would tell Alan his fate).

36. 36. Chris Said：

Editing aid.

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MS Reference Sans Serif 9 point regular
& nbsp; (hard space code no space after & but do include the ; )
<-hard spaces to left. Charmap character 0xA0

Also ALT+0160 (must be on the numeric keypad)
You’ll have to make your own width guide, I can’t post it properly.

37. 37. Rohan Said：

thnx Chris

38. 38. Chris Said：

Not that anyone seems interested. I’ve found a better way out of the dilemma that I had. I can safely say, that from my new realisation, that “the unnamed man” is well-defined after all. (Strangely that upsets me though).

In the case where I said that the warden would only name Bill if he had no choice, I realise that that could only happen in 1 of the 3 possible scenarios – the scenarios are equally likely (until the Warden names Bill). i.e. I had forced the least likely scenario.

39. 39. Chris Said：

To finish. How you interpret the question depends on whether you regard Bill having being named as being a requirement (in which case the conditional probability kicks in) or as an example of what the warden said.

Clearly this alternative interpretation doesn’t seem (to me) to be an issue with the coin flipping problems. Or is that what people are really not understanding? i.e. is their intuition causing them to assume that the coin landing heads up is an example, rather than a requirement?

In the case of the prisoner problem, as posted it was faulty (in view of the answers that were intended); it should have stated that the warden is equally likely to name Bill or Charles when both were going to be executed. That’s my fault. I’d misapplied pre-knowledge of the solution which I had obtained for a variation on the problem (where the prisoner’s name was not specified).

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