Best Of

The probabilities are fixed when the apples go in the bag but you gain information about the bags. That's the key thing.

The information doesn't change the probabilities. That's the key thing.

Which would be fine if the bags were not restricted. But they are. Just because bag a is three random apples, doesn't mean that follows later. The statistics of what you see depend on the evidence you are given.

Nope

Your odds of throwing a 6 don't increase just cos I show you that side of a die

Consider the situation with two apples. Four different bags are GG, GB, BG, BB. If you see an apple from a bag that is G that bag cannot be BB. Therefore it's one of the GG, BG or GB. So your random apple is now 4/6 chance of being G.

The probabilities are fixed when the apples go in the bag but you gain information about the bags. That's the key thing.

The information doesn't change the probabilities. That's the key thing.

But it does!

Eg what if, faced with the bag of 4, you FIVE times select an apple at random (before putting it back) and every time it's a good one.

Is this going to impact your assessment of the mix of apples in that bag?

Of course it is. You'll decide (rightly) that they are probably mainly good.

Yes, but IT DOESN'T CHANGE THE PROBABILITY

IT DOES. Having done 5 random draws of a good apple you are armed with information telling you that the probability of picking a good apple from that bag is very high.

This is not like red/black in roulette or a dice.

It really is

You seem to be arguing that playing Russian Roulette and not getting shot 5 times makes it less likely you will get shot on the 6th attempt...

And remember the original problem is which of 2 bags would you pick.

You know both of them have at least 1 good apple, but knowing that doesn't make 1 bag better than the other

You know more than that.

Clearly you either didn't read the question carefully, or you don't understand probabilities well.

Draw out the set of possible outcomes for both, with their odds, and see if they're the same.

Hint: They're not.

The probabilities are fixed when the apples go in the bag but you gain information about the bags. That's the key thing.

The information doesn't change the probabilities. That's the key thing.

Which would be fine if the bags were not restricted. But they are. Just because bag a is three random apples, doesn't mean that follows later. The statistics of what you see depend on the evidence you are given.

Nope

Your odds of throwing a 6 don't increase just cos I show you that side of a die

That's not an apt comparison.

It really is

The whole point is that the information you are given about the contents of the bags doesn't change the probabilities inside the bag.

Showing you the dice doesn't change the odds.

Ok one last try ...

Imagine you ONE THOUSAND times pick an apple at random from the bag of 4 and every single time it's a good one.

Are you saying this does not change the probabilities of what's inside that bag very strongly in favour of all 4 apples being good?

Course it does. C'mon.

Flipping a coin heads 100 times doesn't change the odds of the next throw.

The odds in the bag are fixed. They don't change no matter how often you pick right. Any right pick doesn't change the odds of the next pick.

But you don't have 1000 picks

There are 2 bags. The odds in each bag are fixed. Which bag do you pick?

No, the choice of apples in the bag is fixed.
But you don't know what it is, only that each apple had a 50/50 chance of being one thing or the other.

By picking one out, and finding it's a good one, you have changed your knowledge. Every other apple still has a 50/50 chance of being either good or bad.

You don't understand the Monty Hall problem.

The probabilities are fixed when the apples go in the bag but you gain information about the bags. That's the key thing.

The information doesn't change the probabilities. That's the key thing.

Which would be fine if the bags were not restricted. But they are. Just because bag a is three random apples, doesn't mean that follows later. The statistics of what you see depend on the evidence you are given.

Nope

Your odds of throwing a 6 don't increase just cos I show you that side of a die

That's not an apt comparison.

It really is

The whole point is that the information you are given about the contents of the bags doesn't change the probabilities inside the bag.

Showing you the dice doesn't change the odds.

Ok one last try ...

Imagine you ONE THOUSAND times pick an apple at random from the bag of 4 and every single time it's a good one.

Are you saying this does not change the probabilities of what's inside that bag very strongly in favour of all 4 apples being good?

Course it does. C'mon.

Flipping a coin heads 100 times doesn't change the odds of the next throw.

The odds in the bag are fixed. They don't change no matter how often you pick right. Any right pick doesn't change the odds of the next pick.

But you don't have 1000 picks

There are 2 bags. The odds in each bag are fixed. Which bag do you pick?

The coin is getting re-tossed each time and it's always 50/50. So each toss tells you nothing about the next one. Assuming not bent of course. Same with dice and roulette.

This otoh is a fixed bag of apples. It's not being reassembled after each random pick and replace. Therefore each random pick and replace does tell you something about its contents. And the more you do the more it tells you.

That's the essential difference.

Plenty of ways you can look at the voting intention of 16-17 year olds, but here's a fun one:

➡️ PE and Business Studies students are much more likely to vote Reform
🌍 The Green Party does best with those who enjoy Drama at school
🔶 Lib Dems do best with modern language fans
https://x.com/edhodgsoned/status/1965428390732976179

Uh oh.

PE and Business Studies.

Tories meanwhile do best amongst those who loved Maths, like Rishi and Geography. Labour do best amongst those who liked English then Maths best

We do appreciate of course that these are just for a bit of fun and are based on tiny subsamples. No more accurate than VI by biscuit choice

Royalists like Bourbons.

While Republicans go for Garibaldi's.

Andrea Jenkyns is partial to a Lincoln biscuit

I would think something half baked would be the Reform choice.

Or a biscuit made from distilled rage and bits of flag

Or perhaps a Madeline in Remembrance of Things Past.

Nice (Treaty) biscuits for the Remainers.

Consider the situation with two apples. Four different bags are GG, GB, BG, BB. If you see an apple from a bag that is G that bag cannot be BB. Therefore it's one of the GG, BG or GB. So your random apple is now 4/6 chance of being G.

Ummm, that's the point. Your odds have not changed because you already have the information when you start.

Let's try this

1 bag with 1 apple. You know 1 apple is good. What are your odds of picking a good apple from that bag?

1 bag with 2 apples. You know 1 apple is good. What are your odds of picking a good apple from that bag?

1 bag with 3 apples. You know 1 apple is good. What are your odds of picking a good apple from that bag?

1 bag with 4 apples. You know 1 apple is good. What are your odds of picking a good apple from that bag?


The original question is about comparative odds between the bags. The odds don't change during the game.

Knowing 1 apple is good changes the comparative odds between the bags at the start of the game, but doesn't change the odds for any given bag during the game.

The start of the game is irrelevant. You don't know what is in the bag beyond the information supplied. If there were a bunch of random three apple bags at least a few would have all bad apples. But we know that our bag is not one of them, so the remaining chances suggest a random apple is more likely good than bad, and better odds the fewer apples there are, as the effect is bigger.
And on that if you can't be persuaded, I'm back to knitting and watching a 0-0 draw...

FPT:

I just saw Concorde. Didn’t know they keep one at Heathrow. But so it is

Kinda poignant

And STILL beautiful

Concorde is proof we were more advance in the 60/70s than we are today.

Utterly incorrect. We have the science and technology to build a Concorde replacement, or something better than Concorde, today.

What we don't have is an economic market for such a plane.

There was barely an economic market for Concorde at the time. London/Paris to New York was the only one that airlines ever found, and I'm not sure about Paris.

And now we have video calls, which do a decent percentage of what Concorde did for a tiny fraction of the cost.

Progress sometimes looks odd.

Concorde and its American oppos were supposed to be the mass market and the 747 was an interim design made to be easily convertible to freight loading when the self-loading cargo all marched off to go on Concorde and the Boeing 2707.

instead it ended up being close to a classic Veblenian luxury item (how close depends how seriously one can take the argument about nipping over to NYC for a quick head to head by captains of industry and bacl in time for dinner).

BA famously did a survey of their customers, asking them how much they thought their own ticket had cost. They way overestimated the actual price, so BA doubled the price of scheduled LHR<>JFK Concorde tickets. A tiny percentage had actually booked it themselves, most being booked by PAs and on behalf of others via agents and management companies.

Two triumphs of engineering kept Concorde flying as long as it did. One was the aeronautics that built it. The other was the financial engineering around the time BA was privatised that kept it going.

(The TLDR is something like- if the costs to date are written off, it only has to fly a couple of hours a day, unlike an easyJet Airbus now. Then you can fly full with a small number of people paying top whack for decent cheese and wine. And then the costs work. There's a brilliant telling of the tale in Francis Spufford's book, Backroom Boys.)

The freedom flotilla drone attack is my favourite story of the day

It seems very clearly to me to be a self-inflicted flare attack

A great example of how an untruth can go around the world before it’s clearly proven to be wrong.

As opposed to the murder of Iryna Zarutska, where the media totallly ignored a cold-blooded and unprovoked killing that took place in public, because it didn’t fit their preferred ‘narrative’’.

It's all over the media now.

Sadly with, in the US at least, some terrible headlines about how drawing attention to these events plays into Trump’s hands about violent crime in cities, and how widespread CCTV “leaks” risk social unrest.

Rather than how someone can be arrested 14 times for violent offences and still be out on the streets.

https://x.com/bennyjohnson/status/1965041387386114196

Was he convicted of any? If not, the number of arrests is irrelevant.

One of the paradoxes of extremely harsh sentences is that juries become reluctant to convict.