/sci/ - Science & Math

>>12196655
Say I have flipped a coin 1000 times.
I know what the results of those flips were, but you don't.
I give you the coin to flip once.
What is the probability it lands heads?
What is the probability it lands tails?

>>12196655
Say I mint a perfect coin, and flip it once, then destroy the coin.
it either landed heads 100% or tails 100%
That coin can't be flipped any more, since i destroyed it
So in the long run, why must it average to 50:50?

>>12196655
>>Inb4 each toss is a separate event untouched by the past
>That doesn't really help me
That should help you, the coin has no memory, flipping it doesn't modify it in any way.

If it did, we'd all be selling trick coins that had landed on tails the last dozen times.

But your main misunderstanding is:
>Since it is a perfect coin the heads and tail will have to even out to 50% in the long run
That's sort of the Law of Large Numbers, but you're missing that the "long run" is infinitely many coin flips. It doesn't force the coin to ever behave differently, it's just the pattern that has to arise if it has those probabilities.

From the moment a coin is minted, it starts acquiring a probability charge. For example, whenever a coin lands heads-up, it gets an aproppriate additional charge of tails-probability.
Obviously this effect is stronger for newly-minted coins. Th is why coins used for professional gambling have to be flipped several thousand times until they can be approximated as unbiased.

>>12196666
hmm, interesting way of putting it and I get why you are saying, but it still doesn't click with me.

I would say it is 50/50 if I didn't know about the 1000 previous tosses, but I do know they were all heads.

What if we did a bet and you put your money on heads, but before I toss the coin I tell you that the previous 1000 tosses was heads and give you the chance to change your bet. What do you do? Keep in mind it is a theoretical and perfect coin and toss

>>12196694

If you flipped a coin 10 times, and got 10 heads in a row, would you expect the 11th to have a higher probability for tails?

What about if i get 2 heads in a row, is the 3rd throw still 50:50?

>>12196655
HHHH has the exact same rarity, and exact same chance of happening, as HHHT. The only difference between HHHH and HHHT is that one has 1 T and the other has 0 T, and any meaning found in this is human selection bias.

Let's say you score a sequence of coin flips by counting the number of heads minus the number of tails.

The average score over all possible outcomes of N flips is 0 (some +, some -, some 0).

The average percent heads is 50%.

Now let's say you start with 1000 heads in a row and then do N more flips. the average score after 1000+N flips will now be 1000

But the fraction of heads will be

(1000 + 0.5N)/(1000 + N)

>Take N->infinity
>50%

>>12196685
>, it's just the pattern that has to arise if it has those probabilities
Well what if we take away the infinity and instead use a large number like 1 quadrillion. Would you say it is more likely that after that amount of tosses the stats are closer to 50/50 than to 90/10?

>>12196694
>What if we did a bet and you put your money on heads, but before I toss the coin I tell you that the previous 1000 tosses was heads and give you the chance to change your bet. What do you do? Keep in mind it is a theoretical and perfect coin and toss
There's no reason to change if the coin and toss are indeed perfect. Each throw is an independent event.

>>12196655
This confused me for years and years too. The best way to understand it is the idea of independent observations. All new observations are independent from the past and the future. That means that if you only look at the next toss, it will still be 50/50.

However, your intuition is not about the next toss but about the sample. It’s highly improbable to have the sample of a fair coin to be heads 1000 in a row. But when we ask, “does this affect the next coin toss?” the answer is no because then it won’t be independent. Think of it this way, if it was affected by a 1000 in row coin toss, that means the coin is somehow able to record and be affected by previous coin tosses.

I think an auxiliary theory that might comfort you lies in cryptography. For a brute force attack with a probability of 1/N, on average, it will take N/2 attempts to guess the passwords. The law of large numbers in statistics will also lead you toward understanding your intuition.

>>12196655
Each coin flip is independent of the previous one, retard.

>>12196722
It only "has to" even out at infinity. It's more likely to be closer to 50/50, but there's no guarantee. A quadrillion heads in a row is improbable, but only as much as any other combination of a quadrillion flips.

Try drawing out a little tree, branching out at each coin flip.
After two flips, you have HH, HT, TH, and TT.
After three flips you have HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.
As you keep going, you'll find that combinations closer to an equal number of heads and tails are the majority. It's not from something messing with the odds (we didn't flip a single real coin), it's just that a random path with equal probabilities will tend to end up near the middle.

If you don't want to work it out by hand, go to https://anydice.com/ and give it something along the lines of "output nd2 - n" (where n is how many coins). The output will be mostly around half of how many coins you asked for (meaning half were heads).

>>12196655
>keep tossing a coin until you get 1000 heads in a row
>your coin is predetermined now
>make a bet with your friend for ez win
does this feel logical to you?

>>12196775
People would be using machines to "load up" their coins with probability.

>>12196773
>It's more likely to be closer to 50/50, but there's no guarantee
If it is more probable then why is it not more probable for the last toss of the 1 quadrillion to be tails if the stats on the tosses before are 10% tails and 90% heads

I am unable to compute this guys. I might have brain damage?

>>12196775
>build robot dice roller
>have it roll d20s non-stop
>sell due to be lucky dice to dnd players
>sell due to be unlucky dice for them to give to their dms

>>12196786
>People would be using machines to "load up" their coins with probability.
Haha, this made me laugh, but also think the most. Still not convinced though.

Besides it would be almost infeasible expensive and time consuming to load a coin with 1000 heads

>>12196655
If a coin was tossed 1000 times and it went heads every time, you'd have to be the dumbest motherfucker on the planet to bet on tails. It's like saying if the Sun rose for the past 1000000 days it won't rise tomorrow.

>>12196655
Let's imagine dependent events and how they differ from independent.
Say you have a box of seven crayons, one for each color of the rainbow. The initial probability of picking red is in E1 P(R) = 1/7. You pick a non-red color and remove it from the box.
Now the probability of picking red is no longer 1/7, but 1/(7-n) as the box of crayons changes in response to the result of a former event.
Such that, in E2, P(R)=1/6, pick and remove any color. E3, P(R)=1/5, 1/4, 1/x, until red is picked.
So imagine now our coin toss. There are only two possible outcomes to any event: H or T (sideways memes excluded). The probability of each is 1/2.
You flip the coin and it lands H. The probability of that outcome was 1/2. You flip it again. It is the same coin, it has two sides, we have not altered it. P(H) still = 1/2.
>but P(H^x) is very improbable as x -> inf!
Yes it is. But the probability of rolling tails is still 1/2 no matter how many times heads lands in a row. The only thing that is changing is the probability of having another sequential heads result.
Such that P(H) = 1/2, P(2H) = 1/4, P(3H) = 1/8, P(nH) = 1/(2^n).
This is also why generally as the number of events -> inf the distribution of outcome tends to be split 50/50.

>>12196791
You're just looking at it in a way where part of the picture is hidden. Try drawing out that tree, at least mentally. Each path in the tree is equally likely to happen. But as you add more coin flips to it, most of them are going to end up with a roughly equal number of heads and tails.

The paths for "all heads" and "all tails" still exist, as do "all heads until a tails" and "all tails until a heads", but as the tree grows the probability of those paths goes to zero, while the balanced-ish outcomes become more common. There's only one way to get all heads, there's a lot of ways to get outcomes evenly split.

If you accept the multiverse then there is a universe in which literally every coin toss comes out heads.

>>12196824
Honestly, I wonder what it would be like to live in a world where nothing appears to be random. I'd imagine religion would be a lot more popular.

>>12196820
>The only thing that is changing is the probability of having another sequential heads result.
How can the probability of rolling another heads be lower without the probability for tails increasing?

>>12196822
Ok lets say you are at the red circle in pic related. You have just tossed your 3d heads. Isn't it more probable that in your next toss you will end up in on of the other 15 branches? To me it looks like it is only 1/15....oh lol I think it finally clicked hehe. God damn it. I just needed to type it out for myself. Thanks a lot for all the great posts guys

>>12196843
>I'd imagine religion would be a lot more popular
Why would that be?

OP the answer is that improbable, anomalous streaks are diluted by the more probable, closer to the expected results the larger the sample size becomes.

Flip a fair coin 10 times: 10 heads, 100% heads.
Flip it 10 more times: 5 heads, 5 tails, 75% heads
Flip it 80 more times: 40 heads, 40 tails, 55% heads
Flip it 900 more times: 450 heads, 450 tails, 50.5% heads

and so on... you see that the overall probability converges on the true probability without tails ever becoming more probable.

>>12196879
Because the outcome each event has an absolute probability of still just 1/2. When I say absolute understand I mean when considering an infinite amount of events.
The reason why in relative terms the probability of an additional consecutive head given n amount of previous heads is lower is because the probability of either outcome is 1/2. So when there are many heads in a row the probability of that trend continuing is gradually lower.
Should the probability of a tails be increasing with each consecutive head, it stands to reason that given H^n outcomes [n -> inf], P(T) -> inf, which cannot be as P(T^n) should approach zero as n -> inf.
In other words having 1000 heads in a row will not lend itself to having 1000 tails in a row starting on the 1001th event.

>>12196927
I'm assuming in the unlucky-verse, macro-scale "random" comes up the same, like coins always landing on heads, etc. Dice always coming up the same, no matter how you weigh them would be very hard to explain with physics that didn't include some consciousness behind the scenes.

>>12196943
Thanks. This was a good way to explain it

>>12196804
Holy shit you're a schizo

>all these people falling for it
is it still summer?

If you get 1000 heads in a row its obviously almost 100% chance the next ones gonna be tails.
To prove it just calculate the probability of getting 1001 heads in a row to see how minuscule it is.

>>12197143
this, everyone else in the thread is trolling you OP
by the way I'm opening a casino, care to play a few games? I'll give you good odds

>>12196733
In ideal mathematical context thats true, but in reality each cointoss obviously affects the physical state of the coin and the environment so to alter the probabilities of the next toss a bit.
Now imagine tossing 1000 heads in a row. Obviously the probability that the next one is also heads is minuscule, even though mathematically it's still 1/2.

Its essentially the same as monty hall. In monty hall, if you view your choices as independent then the probability will be 1/2. But your choices are obviously dependent and in actuality the probabilities are 1/3 and 2/3. With hundred doors theyre 1/100 and 99/100.
Same with the coin tosses. If you assume the tosses are independent you will get probability 1/2, while in reality where they coin tosses are dependent you get something like 1/2^1000

>>12196952
>>12196987
>>12197143
>>12197147
>>12197150
>>12197166
WOW AWESOME POST KIND STRANGERS

Here, have some gold and an upboat for the great post ;) you deserve it!

>>12197150
In the real world, if you toss 1000 heads in a row, the probability that the next toss will also be a head, if you toss it the same say, is pretty high.

>>12196655
It is only more likely the next toss is tails in the context of an arbitrary number of trials.
The probability of flipping 1000 heads and then flipping a 1001st heads is unlikely only if you are considering what would happen if you flipped 1001 coins some arbitrary number of times and then analyzed your results.

The reason it is unlikely to get heads five times in a row is that you are rolling a 50% chance 5 times, and there are so many possible results from those rolls that HHHHH is only (0.5)^5 = ~3% of the possible outcomes.
Every single time you flip the coin the chance is 50/50. The "likeliness" of rolling multiple heads in a row only goes down because each one is a 50% chance.

>>12197178
Imagine having this image saved to your computer. I would probably kill myself out of shame.

>>12197096
It's always summer in coronaland

>Imagine having this image saved to your computer. I would probably kill myself out of shame.

>>12196694
I mean if the previous 1000 tosses were heads that tells me the coin probably isn't fair. I'd be happy to stay on heads.

But that's besides the point and you pose a fun question.

>>12196655
Gentleman, I think you're misunderstanding the kinds of probability, let me explain.
You think that the next toss woud be more probable, because you're thinking about the whole sequence, and there you're right, although all sequences have the same probability, the distribution of heads and tails in them follows a binomial distribution, So they tend to be 50/50.
But when you're talking about the coin per se, the probability is 50%.


tl;dr: Macro isn't micro.

>>12196655
Was the coin tossed before you started the 1000 times streak? Or is it fresh off the mint?
You flip and end up with 1000 heads. You put the coin down, and walk away.
At what point does the "magic" of luck wear off? 2 hours? 1 day? 2 months? When, if you pick the coin back up and flip it, does the probability change? Does the probability change over time? If you flipped 1000 heads, and your coin gets dropped, and is lost for 1000 years. A random person finds it. Is it "loaded" for heads?
You find a coin in the gutter. Does it matter how many times it was flipped before if you don't know them? Do you wait, and if you wait, does the probability change?
You find a greek coin in the dirt. Does it matter whether it was flipped before? For X amount of time? If you flip it, does it 100% depend on all flips before it?
What counts as a flip? If it falls out of your pocket, is it flipped? If you flip it, don't record the answer, and flip it again, does the "hidden flip" determine what the next flip is?
What is a flip?
Is it when you record it on paper?
If it turns 2 times in the air, is it a flip?
3 times?
17 times?