[HWR_14]

> the mere act of doing it more often dramatically changes the risk.

Kind of. However, you already know that the first N outings didn't have a disaster. So those should be discarded from your analysis.

Doing it N times more has a lot of risk, doing it the N+1th time has barely any.

[323]

This is called the Turkey fallacy: the turkey was feed by humans for 1000 days, and after each feed event he updated his belief that humans care for him until it's now almost a statistical certainty.

[lapetitejort]

Is this the reverse of the Gambler's Fallacy? Instead of "The numbers haven't hit in a while, therefore they're going to hit soon." it's "The numbers haven't hit yet, therefore they're never gonna hit."

[jaggederest]

Also known as complacency. Working in a woodshop, one of the things you are most vulnerable to is failing to respect the danger you're in. This is why many experienced woodworkers have been injured by e.g. a table saw - you stop being as careful after such long exposure.

[323]

A related thing is normalization of deviance. You start removing safety because you see nothing bad happened before, until you are at a point where almost no safety rules are respected anymore. You can see this a lot in construction videos.

[toss1]

Yup, complacency can kill you.

In this case [0], a skydiver forgot to put on his parachute...

https://reverentialramblings.com/2018/08/15/the-skydiver-who...

[stavros]

Oh man, that's terrible. I can certainly understand how someone without a checklist that is verified by two people can do that, especially if you have a backpack on to mask the fact that the parachute is missing.

Many times if I wear a tight jacket in the car, I forget to put my seat belt on, because I unconsciously mistake the pressure of the jacket for the seatbelt's, even though putting on a seat belt is usually the first thing I do.

Poor guy.

[ArcMex]

I generally take off my jacket before driving for that very reason.

[ask_b123]

Wow, that's terrifying and a good cautionary tale.

Also, when I read

> I’m hoping you can you forgive me as a minister of religion for likening this story to a spiritual cautionary tale. Yes, we do need to live each day as if it might be our last.

I thought, "Hmm, sounds adventist", and sure enough :-)

[blunte]

And why pilots traditionally work from checklists, even if they've done the process thousands of times.

[ska]

That only applies if you are updating priors. In this case the odds are fixed, the GP is correct.

[323]

The odds of a rocking accident are known and fixed?

[ska]

Probably not. But they aren't affected by the previous N climbs, at least as described by GP post. They are considering a fixed odds event, and the probability of (bad thing happens) over a sample path through time. That's not the turkey fallacy.

In other words , the difference between the turkey and the climber is the climber knows the odds (at least nominally) , and it’s important .

[withinboredom]

All this reminds me of “if you are immortal and cannot be harmed, what are the odds of getting ‘stuck’?” I’d venture 100%.

[totetsu]

Surely sometime about the turkey getting fatter each time complicates this example.

[8note]

hows yesterday's tree impacting today's?

[pessimizer]

If you'll die if a roll of three dice comes up sixes, you're not really in a lot of danger. If you do it every day, you have about 15 months to live.

[jrochkind1]

If you've already done it for 12 months without it happening though, the next 3 months are no more dangerous for you than for someone starting from scratch.

[MattPalmer1086]

Very true. The only winning move is not to play!

[tshaddox]

That's true, but usually when we are deciding which actions to take, we're not comparing "I take actionA" versus "I take actionB," rather than comparing "I take actionA" versus "some random other person takes actionA."

[jrochkind1]

OK, the next 3 months are no more dangerous for you than if you hadn't spent the last 12 months doing it. What you did in the past has no bearing on the chances going forward. I'm not sure if it's more clear to say it like that or not. Clearly, humans have a lot of trouble speaking and thinking clearly about statistics.

The next three months are no riskier than your first three months were. They don't become more risky because they will add up to 15 months total -- once you've already finished the first 12 without incident.

[kenjackson]

For the dice roll example that is true. But other examples it isn’t. For example the MTBF of a device that has run for x hours approaching the MTBF is probably more likely to fall in the next x hours. Or if there is some cyclic behavior. Like waiting outside for a hot day.

[moralestapia]

>you have about 15 months to live

Or a few minutes ... or 20 years.

That's the thing w/ statistically independent trials.

[ska]

That's like the difference between

You could win 100mm in the lottery (true statement!)

Lottery tickets are a good investment (almost always, false statement).

Planning on "well it could happen, technically" isn't a good approach.

[blunte]

But when looking at a possible positive outcome, such as the lottery case, it can "make sense" to buy one ticket.

Your chance of winning goes from No Chance to A Chance, which is an infinite improvement.

[ska]

That's not how this works as a rational investment choice.

It's true that you can never win a lottery you don't enter, but the expected value of that ticket is vastly lower than what you paid for it. That means, as an investment, your $10 will be expected to do better in literally anything with a positive return.

If you are buying > $10 worth of dreaming (for you), fine - but that's consumption.

[barrenko]

Yup. Don't do stuff (repeatedly) that have an absorbing barrier - https://medium.com/ml-everything/nassim-taleb-absorbent-barr...

[QuercusMax]

Anyone who's rolled double natural 1s with advantage would never take this bet - and your example is twice as likely to occur!

[Someone]

The expected value will be 6³ = 216 days or about 7 months. Where do you get the factor of two from?

Also, “not really in a lot of danger”? Those odds are worse than that of a 100 year old in the USA (they have a life expectancy of over two years)

Certainly, as an additional risk, it’s high.

[wizzwizz4]

You forget: once you roll three 6s in a row, you're dead, and you don't roll any more. Your expected calculation assumes that people keep rolling after they get 666.

Though I'm not sure where they got their figure from, because there isn't an “expected time to live”; there's a 90% probability to live time, a 5% probability to live time…

[Someone]

There’s a difference between expected value of number of days you’ll survive and the number of days a given fraction of the subjects will survive, but I don’t see either supporting the claim “If you do it every day, you have about 15 months to live”.

  (215/216)^450 ≈ 0.124

, so about one in eight will survive for 15 months or more. The “5% probability to live” time is around day 645 (about 1¾ years):

  (215/216)^645 ≈ 0.0501

the “half will survive at least for” point is around 5 months:

  (215/216)^149 ≈ 0.501

[jim-jim-jim]

Funny book recommendation: "The Dice Man" by Luke Rhinehart.

[cipheredStones]

If you think a 1/216 chance of sudden death isn't a lot of danger, I don't want to go rock climbing with you!

[withinboredom]

Don’t look at the actuarial tables. The odds are worse than that over a year’s time after ~35

[sin7]

What am I missing? 6 x 6 X 6 = 216 or about 7 months.

[d1sxeyes]

RandomSwede's comment is accurate, but maybe the below can help add some 'flesh' to their response.

Basically, the problem is that you can't just multiply it all together.

(1/6) ^ 3 is correct, and the probability of rolling 3 sixes is indeed 1/216 today, but if you repeat independent events, you don't just add up the probability.

Imagine instead of dice it's coins, and it's only two. Your odds of getting HH today are 1/4, but the odds of getting HH by day four are not now 4/4. We know that it's possible, although unlikely, you could flip coins for the rest of your life and NEVER get two heads. So we know that you can't ever have odds of 4/4 (or 1), only odds that approach 1. So that means that we can't say 216 days from now will be 216/216.

Instead, you need to work out the probability of the event NOT happening, and then repeatedly NOT happening independently (so we can multiply together to get the probability.

For our four coins, the probability of NOT getting HH is 3/4. On Day 2, the probability of NOT getting HH on both occasions will be (3/4)×(3/4), (9/16, 56.25%). By day 3, it will be (3/4) × (3/4) × (3/4), or 27/64. On day 4, it'll be 81/256, or 31.6%. Now we can subtract from 1, to work out that by day 4, the odds of us having hit HH are almost 70%.

As RandomSwede explains, there's a 50% chance that you will have rolled three sixes by day 149. By day 496, you're down to 10%.

[sin7]

    runs <- 10000
    x <- vector(mode = "numeric", length = runs)
    for (i in 1:runs){
      while (sum(sample(1:6, size = 3, replace = TRUE)) != 18){
        x[i] <- x[i] + 1
      }
    }

    summary(x)
    quantile(x, c(0.5, 0.8, 0.9)) 

    > summary(x)
       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
        0.0    62.0   149.0   216.2   300.0  1902.0
    > quantile(x, c(0.5, 0.8, 0.9))
    50% 80% 90%
    149 350 495

A simple simulation. Run 10K times. Count the number of times it takes for three dice to add up 18.

The numbers very much agree with you. The median is 149. The 90th is 495 in the simulation, which is close enough to 496. There is very much a long tail in the data. So, the median and the average will not be the same. Is it a coincidence that mean is a 216?

[d1sxeyes]

No, I don't think this is a coincidence, but I'm not completely confident in saying that.

Thinking about it doesn't make me feel like I'm solving a maths problem. I start stacking ideas and concepts in a way which makes me feel like I'm overlaying them in a way which is incorrect.

It makes me feel like I'm solving a riddle, which hints to me that maybe it's actually a question of semantics and definitions rather than a maths problem.

[randomswede]

Dice (typically) do not have a memory, so whatever happened yesterday will not influence what happens today. If you roll it daily, your chance of surviving at least N days is (215/216)^N, for the specific case of "rolling three 6 on three 6-sided dice" that puts you at ~50% at 149 days and at ~10% at 496 days.

At sufficient scale, even incredibly unlikely things become quite probable.

[sin7]

    runs <- 10000
    x <- vector(mode = "numeric", length = runs)
    for (i in 1:runs){
      while (sum(sample(1:6, size = 3, replace = TRUE)) != 18){
        x[i] <- x[i] + 1
      }
    }

    summary(x)
    quantile(x, c(0.5, 0.8, 0.9)) 

    > summary(x)
       Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
        0.0    62.0   149.0   216.2   300.0  1902.0
    > quantile(x, c(0.5, 0.8, 0.9))
    50% 80% 90%
    149 350 495

A simple simulation. Run 10K times. Count the number of times it takes for three dice to add up 18.

The numbers very much agree with you. The median is 149. The 90th is 495 in the simulation, which is close enough to 496. There is very much a long tail in the data. So, the median and the average will not be the same. Is it a coincidence that mean is a 216?

[randomswede]

Off the top of my head, I don't know. It MAY be related to the fact that 6*3 is 216, but I don't have deep enough statistics knowledge to say for sure. You coudl try it again with 3 8-sided dice and rolling 24, that should give you ~50% at 344 iterations, and ~90% at 1177 iterations. If my supposition that the mean is related to the possible rolls, then the mean should end up being 512.

Iteration counts gathered with Python and a (manual) binary search (actually faster than writing code).

[sin7]

    runs <- 100000
    x <- vector(mode = "numeric", length = runs)
    for (i in 1:runs){
      while (sum(sample(1:8, size = 3, replace = TRUE)) != 24){
        x[i] <- x[i] + 1
      }
    }

    summary(x)
     Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
      0.0   146.0   353.0   511.8   708.0  5112.0 

    quantile(x, c(0.5, 0.8, 0.9))
     50%  80%  90% 
     353  824 1187

Strangely enough the mean agrees. The other ntiles are off a bit, but that's randomness for you.

[montebicyclelo]

The parent comment talks about scaling back the amount of rock climbing they do in order to reduce risk.. And now you are saying that they should go one more time, because a single climb is low risk?

[HWR_14]

Yes. I am saying their analysis of risk is incorrect, and therefore if that's the only reason they aren't climbing then they should climb more often.

[mort96]

I think you're reading it wrong.

After a long life of rock climbing, there's no significant risk of doing it one last time or 10 last times (ignoring the effect of old age itself and whatever).

But when you're in earlier stages of your life, you're asking a different question: You're asking, is this something I want to do hundreds or thousands of times in my life, knowing that each of those times has a small chance of ending my life? This becomes a completely different question.

If I'm 35, maybe I will climb 30 times per year on average for 30 years until I'm 65. That's 900 climbs in total. If my goal is to not die or experience serious injury from rock climbing even once in my life, I have to consider the chance that any one of those 900 climbs will result in serious injury or death. I don't know the numbers for the risks involved, but it seems reasonable to be cautious.

Maybe I don't want to give up on rock climbing altogether, but maybe I can scale it back. If I limit myself to 1 climb per year, that's 30 climbs in total. Much lower risk than with 900 climbs.

This is not a logical fallacy.

[HWR_14]

That would be a reason to have not climbed more than a specific rate ever. It wouldn't be a reason to scale down the rate of climbing as you age.

[mort96]

You're making it sound like it's a decision they made when they got into rock climbing initially, that they would climb frequently while young and then scale back as they get older.

Now, making that decision at the outset does make sense, because it will drastically reduce the number of climbs you make in your life compared to climbing frequently throughout your life, and rock climbing while young is less risky than rock climbing while old.

But importantly, I don't think that's what GP did. It sounds to me like GP spent their youth climbing a lot without considering their mortality, but then decided to scale back because they realized climbing that often for the rest of their life would be dangerous. Maybe they spent the time from 20 to 35 climbing 30 times per year, in keeping with my earlier example. That means they've already climbed 450 times. Risky, but they made it through alive. At 35, they start to consider their own mortality, and they have the choice between climbing 900 more times by keeping to their current rate, and climbing 30 more times by reducing their rate (or something in between). Deciding to scale back makes sense.

There is no logical fallacy.

[morgante]

The assumption is that it's desirable to have a descending climbing frequency instead of uniform.

This makes a lot of sense, as when you're younger frequent climbing would help you to develop proficiency quickly and your body allows you to joy it fully. Plus the social benefits are probably higher when younger.

Once you're older, it's potentially less enjoyable (as your body ages) and you don't need to worry as much about rapidly gaining proficiency.

[digital-cygnet]

I think what you're missing is that they are not avoiding "going rock climbing one more time"; they are avoiding "being a person who habitually rock climbs", because while each excursion is low-risk the aggregate effect will be high risk. It's like smoking -- one cigarette won't appreciably impact your health, but "being a smoker" will.

None of this intended to cast aspersions on rock climbing in particular, just pointing out that a reasonable person, understanding independence of events and not falling prey to any fallacy, could reasonably make this decision based on their personal risk tolerance

[ummonk]

Yes, or more accurately there is a frequency of climbing outings at which the marginal increase in satisfaction from an extra climbing is no longer sufficient to justify the increased risk.

[MattPalmer1086]

I disagree, their analysis is perfectly correct.

The more frequently you take a risk, the greater the chance that risk materialises.

Parent wants to lower their overall risk, but doesn't want to stop climbing entirely. So they climb less often.

[landryraccoon]

I don't know how you can make this claim objectively without knowing that individual's preferences.

If an individual decides their risk tolerance is that they will not accept a one in a million chance of injury from rock climbing, how is their analysis incorrect?

[s1artibartfast]

I think it the argument is to make a lifetime risk assessment, opposed to an individual event risk assessment.

If your tolerance is X% death/life, you can calculate the climbing frequency that falls below the threshold.

On the plus side, if you assume the events are independent, you can recalculate and increase the frequency after each climb.

[NineStarPoint]

The point is that they're changing their habits. Of course we ignore the n times they've gone before, now instead of their habits meaning they'd go m more times in the future, they're going to be going p times in the future for some p that is much less than m.

So it's not about how often they've done it over their lifetime so far, but about how many times they will be doing it over the rest of their life.

[srcreigh]

Under this assumption, by the principal of mathematical induction, you can easily do it K more times for any K without taking on barely any risk at each step of the way.

[HPsquared]

The "slippery slope" principle applies here though: N+1 enables N+2, which enables N+3 and so on.

[mkolodny]

Slippery slope is a fallacy, not a principle. Just because you took N steps, that doesn't necessarily mean you will take N+1 steps.

It's a convincing fallacy because sometimes you do take N+1 steps. But just like in the article, heuristics aren't always right.

[HPsquared]

When accounting for human psychology it does have validity: doing an enjoyable activity "one more time" has a risk of a habit forming, which has a non-zero probability. It is indeed possible.

The argument can certainly be used in a fallacious manner (e.g. by greatly exaggerating the probability of the further steps, saying they are inevitable if the first step is taken, etc.). It's logically valid to say that the first step enables subsequent steps to be taken.

Edit: I'd say that the slippery slope is perfectly valid rule of thumb in a lot of 'adversarial' situations. Once one side makes an error or fails somehow, the balance between the two sides can be disrupted leading to one 'side' gaining momentum. Just as between people, a similar 'adversarial' process can occur within the minds of individuals: between two ideas or patterns of thought/behaviour, one idea can gain momentum after a decision has been reached. Precedence is a strong force.

[vojvod]

Slippery slope arguments aren't inherently fallacious. If you can justify one more climb on the grounds that probability of injury or death is very low then you will be able to justify every subsequent climb on the same basis.

[bear8642]

>If you can justify one more …

Reminds me of Terry Pratchett quote "No excuses. No excuses at all. Once you had a good excuse, you opened the door to bad excuses.”

Full quote is fifth here: <https://www.goodreads.com/work/quotes/819104-thud>

[mkolodny]

Slippery slope arguments are inherently fallacies. They don't prove that something will happen.

Just because you can justify the next climb on the same basis, that doesn't mean you will. You could decide that you've already tested the odds one too many times.

[bhuber]

Don't get on that greased sliding board that ends at the top of a cliff. Once you start sliding, it will be hard to stop because of the grease, and then once you slide off then end you will fall and die.

Do you really think this slippery slope argument is a fallacy? FWIW, wikipedia acknowledges slippery slope can be a legit argument when the slope, and it's chain of consequences, are actually real. https://en.m.wikipedia.org/wiki/Slippery_slope . Indeed, this is the very basis of mathematical induction.

[mkolodny]

From your linked article:

> The fallacious sense of "slippery slope" is often used synonymously with continuum fallacy, in that it ignores the possibility of middle ground and assumes a discrete transition from category A to category B. In this sense, it constitutes an informal fallacy.

"If you take N steps, you will take N+1 steps" is a fallacy whenever it's possible that you won't take N+1 steps.

[vojvod]

"You could decide that you've already tested the odds one too many times" was the original point. Someone responded that the N previous times don't matter and N + 1 has barely any risk. Another poster countered that that argument as stated applies not just for N + 1 but for (N + 1) + 1 etc and therefore the slippery slope principle applies.

Of course if you add in "you could decide that you've already tested the odds one too many times" then it's a fallacy to invoke slippery slope because an off-ramp is explicitly specified. In this case slippery slope was mentioned only because N was dismissed as irrelevant.

[afiori]

A pet peeve of mine is that the slippery slope fallacy can be defined as "modus ponens but wrong".

A fallacy should be a incorrect shape of an argument, a incorrect reasoning, not just a false statement.

[TheSpiceIsLife]

Like all fallacies, it's only a fallacy when it's fallacious.

Otherwise, it's just a regular d argument.

[HPsquared]

Maybe fallacies could be renamed "logical hazards" or something like that. Arguments that are at high risk of being false and require extra care, but not automatically false.

[aaronblohowiak]

but the risk is independent. so once you do the N+1 time safely, you are back to N and your next time is _also_ just an N+1.

[twobitshifter]

True but it would be incorrect to assume that you can safely keep basejumping every day in a year, just because you haven’t died in the last 50 days. Eventually the stats say you will be 87% likely to have an accident when you consider your choice at the beginning of the year. It might be day 20 or day 300, but you won’t know what case you end up in. The chance of your next jump being your last is always the same, but that doesn’t decrease the risk of repeated trials.

[jrochkind1]

Not exactly. If you've done it 50 days without an accident, your current chances of the accident happening in the remainder of the year are NOW less than 87%.

If you've made it Jan 1 to July 1 months without an accident, the chances of you making it to Dec 31 are now better than they were on Jan 1 -- because now they are just the chances of you making it six months, not a year.

The chances of flipping 6 heads in a row are 1/64. But if I've already flipped 3 in a row... the chances of flipping three _more_ heads in a row is 1/8, the same as always for flipping 3 heads in a row. The ones that already happened don't effect your future chances.

[twobitshifter]

I meant to say starting a new year after the 50 past days, I see that wasn’t clear though.

[majormajor]

Yes, but when you make a plan to find an acceptable cumulative future risk, planning to do it once a week for the rest of your life is planning to expose yourself to significantly more risk than doing it twice a year for the rest of your life.

You might still die in one of the next 20 instances. But you've added a lot more not-dead time in between them!

Saying "I can do one more with minimal added risk" every single time after not dying is true and yet pointless, because it's not a given that "minimal added risk" = "not dying." It's survivorship bias to not think frequency doesn't affect the cumulative odds of your future planning solely because you've already done a lot of trials.

[Stronico]

Risk is independent of prior events, habits are not - I think that was what the anthropologist story is about

[ummonk]

The risk is independent but the marginal enjoyment isn't. You don't get double the satisfaction from climbing twice as much.

[pessimizer]

Continuing to do something regularly doesn't ever mean you're just going to do it once more.

[nefitty]

Psychologically, behaving in a certain way makes it more likely that you'll behave in the same way in the future. That's an integral idea underpinning justice systems.

[jvanderbot]

In a skill-based game, N+1 has less incremental risk than adding 1 more trial with N-1 games did.

[mrow84]

This assumes a lot about the underlying process, particularly independence. Whilst assuming independence might hold reasonably well for low numbers of samples, the assumption might be increasingly (and dangerously) misleading. The intuition expressed by GP captures that.