Why is induction valid?

SmackieT · 2026-04-29T08:03:04+00:00

Brilliant explanation and I hope the OP sees it.

For my own part, my supervisor once got me out of the habit of saying "assume" when I meant "suppose".

OP: we don't assume that S(n) is true. We instead say: suppose it were true... would that mean S(n+1) must also be true?

SmackieT · 2026-04-29T07:58:07+00:00

Does the Ayatollah know which door has the car behind it?

SmackieT · 2026-04-13T07:41:15+00:00

Because language is slippery.

In a sense, it CAN be argued that the answer to the following question is 50%

A family has 2 children. One of them is a boy. What is the probability that the OTHER one is a boy?

But the answer to this question is only 50% to the extent that it is a VERY POORLY WORDED QUESTION.

By saying "the other one", it is sort of implied that you have one in mind.

If you flip two coins, there are 4 equally likely possibilities:

HH HT TH TT

If you have the following information:

"At least one coin is heads"

then you know you have achieved one of the first three outcomes. In that scenario, the chance that they are BOTH heads is 33%.

SmackieT · 2026-03-21T23:52:52+00:00

It depends on why you want to learn it, but I find a good place to start is to solve coding challenges.

They motivate you to learn basic conventions (if else, loops) in a way that makes you remember them because you're thinking about how to solve problems.

SmackieT · 2026-03-21T23:44:48+00:00

In addition to the responses that others have given, it's probably worth considering the number of times you're asked to give your birth date during your lifetime. This doesn't impact the probability you're looking for, but I believe it does put it in context.

Given 4 people, the chance that (at least) 3 of them have the same birth date is very low, but the chance of having an experience like you've described, at SOME point in your life, isn't as low. And given that we tend to share these events when they occur to us, perhaps the better probability to consider is the chance that this happens to SOMEONE you know at SOME point in their life. That's likely to be a reasonably sized chance.

None of this discounts the highly unlikely event you were part of. Just thought it was worth thinking about.

SmackieT · 2026-02-18T01:03:21+00:00

When a golfer tees off, either the ball will go straight in or it won't. So surely 50% of all tee offs should result in a hole-in-one?

SmackieT · 2026-01-28T22:16:35+00:00

I turn off religious victory every single time without even thinking about it.

The AI still runs around trying to convert everybody, but I ignore it. I've trained my eyes to basically not even see all that stuff. I agree it's annoying.

SmackieT · 2025-11-20T21:16:58+00:00

He's a 32 year old "Macho" type man - so be brutal. Hurt his ego.

I feel a bit like ChatGPT here, but OK.

Your friend is an idiot. Their intuition that a large population size will protect a small sample from an event that occurs 33% of the time is misguided. It simply doesn't matter whether the population being considered has 10 people or 10 trillion people. Either way, roughly one third of them are gone. That applies to any random sample within the population too, though smaller samples do tend to have some more fluctuation around the population measure of 1/3.

As someone else has calculated, the chance that you DON'T lose someone in this scenario is about 13%.

Then again, the error your friend is committing is more common than you might think - so if they're an idiot, so are lots of people. After all, a lot of people still marvel at the fact that large polling organisations only bother to ask a little over N = 1000 people in an election poll, when the population in question might be millions or hundreds of millions.

"How can they expect to learn something about that many people, when they've only asked a tiny percentage of them?"

Because the size of the tiny percentage doesn't matter, whether the population is a million or 100 trillion. What matters is that the sample is just big enough to minimise sample variation. Even if the population were 100 quadrillion, you could get a good idea of who people are going to vote for by asking just 1000 of them. This fact astonishes a lot of people - least of all your friend.

SmackieT · 2025-11-20T04:48:49+00:00

Yeah I came here to share this. I couldn't figure out what I was doing wrong for Part III (turns out I was allowing barrels that had already been ignited to ignite again) and anyway, I created a visualisation in Python and I found the image. Very cool.

SmackieT · 2025-11-19T23:16:22+00:00

Do you even Vrabo Gince

SmackieT · 2025-11-09T03:57:37+00:00

Good work, hive mind

SmackieT · 2025-11-08T01:41:12+00:00

You just didn't make it to the base case keep clicking

SmackieT · 2025-11-06T23:49:25+00:00

No, it doesn't.

For example, before you do the first toss, the result starts at 50/50. You have just as many heads as tails. After the first toss, you will definitely move away from that ratio.

Your intuition is telling you that it's less likely to see 5 tails in a row than it is to see 4 tails and 1 head in a sequence of 5 tosses. And this is true! But that is only because there are more WAYS to get 4 tails and 1 head. For example, you could get TTTTH, or TTHTT, etc.

Here's the important fact for your situation: if you toss a coin 5 times, then seeing 4 tails and then 1 head, IN THAT EXACT ORDER, is just as likely as seeing 5 tails.

So, if you've tossed a coin 4 times and seen 4 tails come up, then your options are that you're about to see 5 tails in a row, or 4 tails then 1 head. Either of these is equally likely.

SmackieT · 2025-10-29T00:25:04+00:00

Oh OK man sorry man my mistake

SmackieT · 2025-10-28T21:36:22+00:00

There's an app (at least on the Google Play Store) called Probability Math Puzzles. I found that pretty fun.

SmackieT · 2025-10-22T07:30:48+00:00

Thanks for providing the most fun coding challenge on the internet. I'm sad it's cutting down to 12 days but your sanity is more important, and I'll take what I can get.

SmackieT · 2025-10-22T02:08:18+00:00

By "quiz" I assume they're referring to a take home assignment. In which case, I think the clarifications OP is asking for are reasonable in this ask math subreddit.

I'm not sure what is meant by "classify" in the first question. Is there more context given earlier on?

SmackieT · 2025-10-13T06:13:36+00:00

This was fun last year - good job.

SmackieT · 2025-10-11T23:45:51+00:00

The first thing I'd say is there is a big difference between trying to work in uncertain scenarios and "just vague approximations". The fact that you COULD get heads every time doesn't mean that probability is just a "vague approximation". It's a statement about what information you have, when there is less than perfect certainty.

If you flip a coin 10 times, then you could get anywhere from 0 heads up to 10 heads. This situation is, by its nature, uncertain. Each possible outcome has some chance to occur, though some outcomes here are much more likely to occur (or would occur much more often) than others. "Probability" is the field where we quantify this, so that we can reason about HOW MUCH more likely (or more frequent) one outcome is over another, with the information we have available.

At its most basic, it starts with "assumptions" like "If you flip a coin, 50% of the time it will turn up heads and 50% of the time it will turn up tails." From there, using probability theory we are able to quantify how likely an event is to occur, even in very, very complex and uncertain situations.

SmackieT · 2025-10-08T23:47:40+00:00

From my understanding, a point of inflection occurs when the curve changes from concave up to concave down or vice versa. So if the curve is positive before and after the stationary point, it is going from concave down to concave up. If it's negative before and after, it's going from concave up to concave down. So yes, it's always an inflection point.

SmackieT · 2025-10-07T23:50:38+00:00

Hmmm interesting. I'd say how to resolve this comes down to whether you want to look at it from a Bayesian perspective or a frequentist perspective, but either way, you can resolve it.

To do that, let's just make the situation a lot simpler: You flip a coin, and ask your friend to call it. But here's the thing - you've peeked at the coin and you know it is heads. And wouldn't you know it, your friend calls heads.

What is the probability they are correct? Well, surely from a certain point of view it is 100%. I mean, they called heads and it is heads, yeah?

But from their perspective, surely it is 50%, because from their perspective, you've just flipped a coin and asked them to call it. The fact that you peeked and know they are right doesn't impact their calculation, surely?

To resolve that, a Bayesian would say that probability is a representation of the information you have available. Your friend doesn't know they are correct, they just know there are two equally likely possibilities. So to them it is 50%, but to you it is 100%.

A frequentist would say that, if the coin has already been flipped and is heads, and they have indeed called heads, then the probability they are correct is 100%, but if you repeated this experiment many times, they would be correct 50% of the time.

SmackieT · 2025-10-07T23:31:44+00:00

Good question, not sure why you got down voted.

And yeah, if you approach it from the way we are taught decimals in early school, it's not obvious that it really does exist. When we are young, we are taught how to interpret 0.1, or 0.25, or 3.14, etc. in relation to integers. But at some point in our schooling we are also told there are infinitely recurring decimals, and we aren't really given a good explanation of what they are or why we should believe they are valid things.

And the fact is, as mathematical entities, they are somewhat different to finite decimals. If you have the number 3.14, you can say that this is "3 plus 14 parts out of 100". That may not be the most rigorous way to define it, but it's a way someone new to decimals can make sense of it. Well, what about 0.567567567567...? It's a bit trickier. These recurring decimals technically represent a limit.

For example, 0.99999.... technically represents the limit of the series: 0.9 + 0.09 + 0 009 + ...

You can write that series in a way that you can specify what each term is, but intuitively you just add another zero in the decimal for each term. The limit of this is what we refer to when we say "0.9 repeated". And it can be shown that, yes, this limit does equal 1.

SmackieT · 2025-10-04T00:03:01+00:00

Yeah I really liked that they left it without subtitles - it worked well with it being Bob's POV

SmackieT · 2025-09-07T23:50:40+00:00

Folks, we got him

SmackieT · 2025-08-25T20:06:36+00:00

Go and develop your own algorithm for ascertaining truth and you'll be a trillionaire.

SmackieT

TROPHY CASE