Paradox of the drunken man

[Izzy-qfdo]

It's probable it's not his house.

[absofjelly]

He's most likely to open the door on the last try. Once it's open he's not going to keep trying. But then he is drunk.

[christopherwellman]

He's also most likely to get it the first try because he always tries the first time.

[ismaeelabuabdallah]

lol the question is tricky:
"On which try is he most likely to open the door?"
Yeah your math is fine.
"On which try is he most likely to find the key?"
Then they are all 1/10.

The difference is that the first one means: "finding the key first"

[mil]

I'm pretty sure a drunken person will fail to open the door even when they select the right key 😁

[AA-]

This is not a paradox! He has a 100% chance of choosing the right key on the LAST try

[LeahyPhoto]

I take 10 minutes to walk to the pub. I take quite a bit longer to walk home: the difference is simply staggering. I have just one key but I can’t get the door into the key for ages.

[nicku]

A woman watches her drunk neighbor trying to unlock his door with a cigarette. - You're not likely to open the door with a cigarette - she says ironically. "Damn, " the guy says, coming to his senses and examining the cigarette in his hand, "that means I have smoked the whole key!"

[hotdogskid]

The key to understanding this paradox is that something can be the most likely while still being not very likely because every other option is even less likely

[myrus]

We usually think of probability in terms of either the expected number of tries (which would be 9 I think) or which ever try has the highest possibility of success (which would be any of them, as they all have a 1/10 shot at working). The paradox I think comes from the less familiar and intuitive question of which try is it most likely to happen on, at least that’s how I got confused on it.

[azaleacolburn]

Your question is poorly phased. He has a 100% chance of opening it on his last try if he gets there.

[BrotherAlpha]

This reminds me of the says, "It's always the last place you look." Of course it is, because once you find it, you stop looking.

[the_real_foamidable]

As a drunken person, I find this most encouraging!

[amadeolopez]

Its in the way the question is asked. The probability of each try is 1/10. But since she is asking on which try is he more likely to get the key right, the answer is the first try, since you would have to get the wrong key to move to the next try.

[Phlebas]

That actually doesn't seem too counterintuitive to me, keeping in mind we're looking at the chance of him finding the keys on any particular iteration.

It's also true that the probability of finding it within, say, the first five attempts (eg. P1 + P2 + P3 + P4 + P5) is going to be much higher than him finding it, specifically, on the sixth attempt.

[hydrolito]

Drunken person still knows what correct key looks like. May have more blurred vision but can still see.

[alphazutn]

For anyone who is confused.

10 coin tosses or dice rolls don't include the 'getting in' part, which is why the outcomes are independent of the outcome of the previous tosses or rolls.

Here, the 'getting in' part means that you can't initiate another completely independant toss or roll, since getting in means not attempting the rest of the trials, a rule which is absent from your normal probability maths class.

This is very similar to the Monty Hall problem, where the idea that doors were excluded changes the way you calculate probability for the final attempt.

You can also do a reductio ad absurdum to better understand this.
Ask yourself: how likely is it that the drunken man will open the door on his 1 trillionth try and NOT BEFORE that? the chances would be really slim, since one of the '1 out of 10' trillion tries before that try should (probably) succeed, the earlier it is, the more likely.

[trigonzobob]

This is why I bring a key ring with just the house key when I go drinking. Makes the probability calculation a lot easier which is important when you're drunk.

[dangerd]

This didn't make sense at first, until I stopped seeing each try as a single event

As a singular event, each key try is the same probability. But as one whole event, it makes more sense that the first try is the most likely to suceed

[kieranlee]

the second key would have a probability of 1/9 as he has removed the chance it was the prior key. this is because he now has 9 keys with an equal chance of opening his door.