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.
In this case [0], a skydiver forgot to put on his parachute...
https://reverentialramblings.com/2018/08/15/the-skydiver-who...
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 :-)
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.
In other words , the difference between the turkey and the climber is the climber knows the odds (at least nominally) , and it’s important .
Or a few minutes ... or 20 years.
That's the thing w/ statistically independent trials.
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.
Your chance of winning goes from No Chance to A Chance, which is an infinite improvement.
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.
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.
At sufficient scale, even incredibly unlikely things become quite probable.
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
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?
Iteration counts gathered with Python and a (manual) binary search (actually faster than writing code).
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
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%.
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
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?
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.
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.
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…
(215/216)^450 ≈ 0.124
(215/216)^645 ≈ 0.0501
(215/216)^149 ≈ 0.501The 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.
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.
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.
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.
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
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.
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?
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.
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.
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.
It's a convincing fallacy because sometimes you do take N+1 steps. But just like in the article, heuristics aren't always right.
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.
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.
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.
> 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.
"Not A -> Not B" is different logic than "A -> B". A is necessary but not sufficient for B.
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>
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.
Otherwise, it's just a regular d argument.
A fallacy should be a incorrect shape of an argument, a incorrect reasoning, not just a false statement.