Suppose you’re on a game show

Off on an airplane in an hour or so and heading to the United States on the way to two conferences. No blogging for a while, and definitely not till I’m on the other side. I therefore thought I’d leave this little puzzle with you before I went.

Suppose you’re on a game show, and you’re given a choice of three doors. Behind one door is a car; behind the other doors, goats. You pick a door – say, No. 1 – and the host, who know what’s behind the doors, opens another door – say, No. 3 – which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

There was an age-thing when I taught this when I first got to Australia those many years ago. Happily I can still see why the right answer is the right one, but I knew already which one was which. Happy to be getting away from local politics for a while and seeing what things are like in the United States as the conventions approach. The phrase “only in America” has deeper meaning than ever but once it was a positive and now there is little about the American way that I can think we’d want to have for ourselves.

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58 Responses to Suppose you’re on a game show

  1. The classic ‘Monty Hall’ problem – even Paul Erdos took some convincing that the right answer was right. 🙂

  2. Bruce of Newcastle

    The guy is a game show host!
    It makes no difference which door you choose because there is no car, only goats.
    And the labels on the three doors are “ALP”, “LNP” and “The Greens”.

  3. It’s to your advantage to switch by just over 17%.

  4. duncanm

    what BoN says

    The host is your friendly assistant at the local polling booth, there are no cars, and you don’t get a chance to change your choice.

  5. Jumpnmcar

    What if the Host is a ” She ” in a white and gold dress ?

    Never thought of that, did ya……

  6. Tel

    The objective is to extract information from the game show host; so we need a behavioral model that would predict his actions. Based on a presumption that the host wants to make the game “more exciting” he would delay revealing the location of the car as long as possible, thus his choice of No. 3 was not a random choice. Had the car been behind No. 3, we predict the host would have opened No. 2 instead.

    This means the answer is “Yes” you would improve your chances of winning by switching your choice to door No. 2.

  7. Shy Ted

    Ahmed says, “I’ll take the goat.”

  8. goatjam

    Whats with all the goat-hate?

  9. Tel

    On the other hand, a different behavioral model about the game show host would be that giving away the car is reducing his profit and he is actively attempting to make you lose. Based on this presumption we believe that if door No. 1 contained a goat, the host would have opened it immediately and kicked you off the show. Thus, the fact that he is attempting to give you the opportunity to switch suggests you should stick on No. 1 and not switch.

    I conclude that more information is required to be confident with the answer. If you had watched the show in the past you would have a pretty good idea how this guy operates, which may involve a mixed strategy on the part of the game show host (e.g. attempting to make the show fit a neat time slot, thus keeping an eye the clock and drawing it out when required).

  10. cuckoo

    Without googling, I dimly recall the controversy over this. A clever woman who wrote a newspaper column (?) gave the correct answer and had to endure all kinds of abuse from pipe-sucking, leather-elbowed eggheads who insisted she was wrong. Gentlemen, start your search engines.

  11. Dianeh

    But if the car is behind door 1 the games show host wants to get you to choose another door and then the whole purpose of showing the goat behind door 3 is to put doubt in your mind and you would pick door 2.

    I think Bruce has the right answer. Too many goats behind closed doors.

  12. Robert Crew

    Off on an airplane in an hour or so heading to the United States

    Me too, hope you enjoy your trip.

  13. Leo G

    Take the goat at door number 3- at least you get a whole goat. Chances are with any other choice that you will win something half goat and half car.

  14. The original choice is 33% likely.

    The second choice is 50% likely. It makes no difference whether you switch or not. Any attempt to gauge the strategy of the host is prone to bluffing since everyone knows the rules and meta-rules of the game, so you are kidding yourself if you think there is a single better solution that the host doesn’t know about and go the other way.

  15. Ah, but I see there is such a thing as conditional probability, and I am wrong. Good puzzle!

  16. .

    This is why I think the monty hall problem is wrong. From you tube comments:

    You are playing Russian Roulette. The shooter has pulled the trigger 4 times, and the hammer has landed on an empty cylinder. Do you think you have a 5/6 chance of dodging the bullet next time??

  17. Leo G

    The original choice is 33% likely.

    Nope. The original choice has now become 50% likely to win a goat, 50% likely to win a car- The same expectation from now choosing Door 2. You don’t know the expectation that the host is trying to help your chances, so there is no advantage to switching your original choice.
    On the other hand, if you switch your vote and win a goat that would look to the audience like there were really three goats..

  18. 2dogs

    The original choice is 33% likely.

    The second choice is 50% likely.

    As always, Monty gets it wrong.

    The player’s first choice means that they never receive any information about what is behind that door, because the host will only reveal one of the other two doors.

    Of the other two doors, the entire 2/3 chance that the car is behind one of the them gets transferred to the door the host doesn’t reveal.

  19. Smarter people than I made the same mistake. It’s a fine dilemma.

  20. Arky

    What if you shuffle the goats after one of the doors is opened?

  21. john malpas

    Elbow the host to one side. open all the doors. Take what you want. Leave.

    The pirate party.

  22. Captain Bastard

    Coldcock the host
    Smash the doors down, throw
    The game show bimbo and the goats in the back.
    Siphon some gas out of the tank, douse the set
    And torch it. Drive away watching it burn in the rear view mirror
    All criminy red and Autumn orange.
    Pick up a couple of VB tallies and hit the freeway
    Heading north.

    Never could stand that show.

    With apologies to Tom Waits.

  23. Uh oh

    Why didn’t the host open door 1 when asked to? Keeping the game going or trying to get the contestant to change picks away from the car? Stick with 1.

  24. T Bull

    Clearly by switching your choice indicates a distinct hatred for government subsidised car industry and you are obviously pro-goats. To remain with your original selection risks being branded an extreme herdist. Goat lives matter.

  25. Uh oh

    Then again maybe he wants him to change picks so he can keep the car for himself. Is fast Eddie the Host?

  26. Tel

    The player’s first choice means that they never receive any information about what is behind that door, because the host will only reveal one of the other two doors.

    Why do you say that? The host may choose to go directly and open up the door of the player’s first choice, does not necessarily have to open any other door than that.

  27. braddles

    Martin Gardner did this one in Scientific American back in the 60s, and had a devil of a time convincing people, even some very smart people, of the right answer.

    One thing to make clear from the original puzzle: the rules are that the host, who knows where the prize is, will never open the contestant’s chosen door, or a door with the prize. That means his choice may be non-random, and so changing to the door that he avoids is a good bet.

  28. Leo G

    Why do you say that?

    There is no outcome from the first trial, so that outcome cannot influence the “second” trial expectation. Conditional probabilities don’t apply.
    Even if you knew from the start that the host was to offer a new choice by removing from the choice one door that concealed a goat, you didn’t know which door that would be, even though the initial expectation of winning the car in that situation would be 50%.

  29. Tel

    There is no outcome from the first trial …

    Yes, there’s an outcome from the first trial. The host could choose to end the game right there (i.e. the fact that you have a second trial at all is an outcome of the first trial).

    Check your Markov chain theory.

    That reminds me, we play a game and flip a coin (government issued fiat currency so you know it is fair) then we write down the outcome (H or T) and then flip again and write down the outcome repeatedly to make a sequence. Any time the last three letters in the sequence are HTT then I win (and the game ends), but any time the last three letters in the sequence are HTH you win (and the game ends). We keep going (sequence gets longer) until someone wins the game.

    How about that one?

  30. J.H.

    Yep, take Door 2 and the new offer…. The reason is, is that when you first chose, you had 1 in 3 chance of winning…. Now you have a 50/50…… Take it.

    If you are going to gamble, always take good odds, not emotion.

  31. Tel

    J.H. if you consistently play that strategy, then as a game show host I could choose to guarantee that you lose every time.

  32. J.H. if you consistently play that strategy, then as a game show host I could choose to guarantee that you lose every time.

    Not true, when you look into it. If you choose Door 1 and the car is behind Door 1, then the host’s choice doesn’t matter and he has no strategy. If the car isn’t Door 1 then the host must choose the remaining door that doesn’t have the car, so he has no choice or strategy.

  33. calli

    Take the goat at door number 3- at least you get a whole goat. Chances are with any other choice that you will win something half goat and half car.

    A Marina!

  34. kevin

    Is the show on before the 6 o’clock news?
    Is the car due to go off?
    Does the host have big tits?

  35. Leo G

    i.e. the fact that you have a second trial at all is an outcome of the first trial

    So given that there was a second trial offered, what was the outcome of the first?
    I can’t see how a sequential state transition process with state-sensitive inputs and state-dependent outputs is useful in the analysis of this simple conceptual problem.
    Your coin flip game can readily be modelled as a toss-synchronous 4-state problem (start/continue, H, HT, stop) where wins are signalled by outputs from the HT state, and with inputs result(T) and result(H). Note that in this model there can be no transitions without a H or T result from the coin toss.

  36. Tel

    If the car isn’t Door 1 then the host must choose the remaining door that doesn’t have the car, so he has no choice or strategy.

    The host is under no such compulsion, the host merely opens door 1 and there’s your goat sir. Game over.

    Ha ha you lose!

  37. TC

    Depends , if you are after a goat , take the option and choose door three .

  38. AP

    Are the goats affirmative action goats with leadership skills?

    That’s all that matters.

  39. Those are not the rules, Tel. Then again, you’re a noted goalpost shifter.

  40. DavidA

    If I ignore the host my chance of picking the car is 1 in 3. The host is simply irrelevant here.

    If moving:

    P(win car) = P(I picked goat) = 2/3.

    Every time I pick a goat – 2/3s of the time – the host reveals the other goat and I move to the car.

    If the host isn’t always picking a goat then the problem is not the usual one which it resembles and the author has screwed up the telling.

  41. Frank

    Another one is the engineers making a 20km length of railway track 1m too long. To fix it they decide to just push it in a bit resulting in a bow upwards. How high does this bulge sit above the ground?

    Approximating the bulge (curve) by the hypotenuse of a right triangle and invoking pythagoras’ theorem gives a height of roughly 100m. This result surprises many mathematicians when they first hear it.

  42. Tel

    Monty you don’t get to invent the rules as you go along. You picked a door, it was a goat, thank you for playing.

    Don’t project your goal post shifting onto others, if you think you are entitled to a second chance then cite the reference where you were promised. Don’t waffle, cite the actual promise made.

  43. Tel

    I can’t see how a sequential state transition process with state-sensitive inputs and state-dependent outputs is useful in the analysis of this simple conceptual problem.

    Because the host can choose to terminate the game immediately and offering you the chance to swap doors is discretionary. Thus, the game show host has revealed some non-random information by the fact that you are given a second chance.

    Look at it this way: you have three choices, and the host also has three choices. That’s a total of 9 combinations. If the host chooses to open the same door as you in round 1 then the game ends immediately. If the host chooses a different door to you in round 1 then it goes onto round 2 and you are down to two choices. Total possible outcomes is 3 + 2 * 6 = 15.

    However, the host is unlikely to deliberately reveal the prize (but no one has specifically ruled that out) and the player is unlikely to deliberately want to lose (but no one can 100% rule that out either). If you remove those unlikely options you still have these possible games:

    * player picks correctly in round 1, host picks from remaining doors, player either swaps or stays.

    * player “does a Monty” in round 1, host terminates the game by opening the chosen door.

    * player “does a Monty” in round 1, host opens the remaining door that does not have the prize, player chooses to swap or stay.

    Total likely options comes to 7 but we need more information about the host in order to know his incentive under the circumstances.

  44. pete of perth

    What if it is a camel?

  45. John Constantine

    Real life approximates of this could be situations like bidding in an auction where just one other person is bidding against you.

    Is his ability to right price the risk better or worse than yours?.

    It is rare that you are well served by repricing your bid just because somebody has offered you information on their price perception.

    ( pump and dump in the share market same thing, also dump to panic and scalp the bottom.)

    Headology says the prizegiver is prepared to swap prizes for entertainment, therefore it well serves you to offer the maximum entertainment product to them, as by becoming a high value goat winner, you increase your chances of getting another chance to play for a goat next time.

  46. 2dogs

    Why do you say that? The host may choose to go directly and open up the door of the player’s first choice, does not necessarily have to open any other door than that.

    Because I read the rules:

    and the host, who know what’s behind the doors, opens another door – say, No. 3 – which has a goat.

  47. John Constantine

    They are going to give you a car or a goat, so these are the nonzero options.

    The game may look like simple odds, but it isn’t about innocently making a one door choice.

    Once the host plays headology with the doors, the simplest way to reset the game is by introduced ing a random probability generator.

    Toss a coin, nothing else except springing for a door and opening it yourself can extract you from operating within the parameters they have set.

  48. Tel

    Because I read the rules:

    You seem to have some comprehension difficulty.

    The quote you gave described one particular example, unless you can point out the bit where the game show host offered a guarantee to always respond the same way in every situation.

  49. Aussiepundit

    Probability theory says that you must switch your choice. This is settled science. Unfortunately only a fraction of the population are smart enough to understand why. This isn’t meant to insult anyone, it’s just how it is.

  50. Chris M

    The door with the car is wider?

  51. Jannie

    Monty, you should have done better. It’s actually known as the Monty Hall problem. It is counterintuitive. But look it up on Wikipedia if you don’t believe it. It’s right sometimes.

  52. Jannie

    Btw, Aussiepundit, these days settled science is not what it used to be.

  53. Grumbles

    Ahh but TeL the question was are you better switching or staying. IF there is such a situation where Host opens selected door to terminate game, then there is no offer to switch and hense it is no longer within the parameters of the question.

    By switching you CHANGE your original 33% chance into a 50% chance. If you don’t change you are sitting on your original 33% chance. It is that simple.

  54. JohnA

    pete of perth #2085662, posted on July 9, 2016, at 2:46 pm

    What if it is a camel?

    Then it must have been designed by the show’s marketing committee!

  55. 2dogs

    The quote you gave described one particular example, unless you can point out the bit where the game show host offered a guarantee to always respond the same way in every situation.

    The problem described appeared to be the ‘Monty Hall’ problem, where the hosts behaviour is how I described it.

    If the rules of the particular show in question isn’t an exactly replica of Monty Hall, and the host is behaving differently, then, granted, one would need to look at the actual rules governing the hosts behaviour for this game. For example, if he always opened another door when the contestant picked correctly, but only, say, 10% of the time when the contestant was wrong, one would never switch.

  56. Bad Samaritan

    OK. Here’s my explanation…..

    You have a 33% chance of picking the Car first up. Two other times out of three (on average), you’ll have a Goat. If you have the Car, you will definitely get a goat by switching. If you have a goat on your first pick there will be one goat left, and one Car. You must switch since the host has shown you where the other goat is. Two times out of three you get the car. One time in three you will be leaving the Car and getting a goat. 66.66% winners. Now back to my first statement….

    You have a 33% chance of picking the Car first up. Two other times out of three (on average), you’ll have a Goat. If you do not switch you will be sticking with the pre-game expected outcomes. 66% Goats instead of 66% Cars.

    The switch wins 66% of the time. The no-switch 33% of the time.

    If anyone here understands how to gamble successfully, you’ll already be using these types of situations to skin punters on Betfair. Cheers.

    BTW. Here’s my comment from 1st July 2016….

    “Bad Samaritan
    #2074603, posted on July 1, 2016 at 9:27 am
    Betfair has it for the LNP. Now $1.13 LNP vs $8 ALP. The actual money is similar, indicating $1.19 vs $6.50. Betfair (ie the punters; it’s a free market) has it about right.

    By contrast, the Betfair US Presidential election is $1.36 Hillary: $4.40 Trump, which has been way out-of-whack for ages, since the actual money indicates it “should” be Hillary $2.05 (slight outsider) and Trump $1.95 ( slight favourite.) This “should be” is now emerging (via the latest Rassmussen poll) to be correct.

    Anyhow. LNP to win.

    The Oz election is tomorrow, whilst the US election is still 4 months away. What this all indicates is that LNP and ALP backers are getting fair value, whilst Hillary backers are none-too-bright; getting an absolute kicking from the Hillary layers (layers bet against something; thinking like bookies).”

    Now guys, odds and probabilities do not mean that raging hot favs must win by huge margins. The odds tell us what “should” happen (theoretically) over time if there were multiple re-runs of the same/similar situation under similar conditions..

    It does not matter, for example, if Djokovic beats Murray 8 times out of 10…but always in tight five setters, rather than in easy straight set matches. What counts is that he wins 80% of the time (ie is a $1.25 chance) against Murray. Close enough is not good enough in sports or in politics..

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