Out of Pod Experience

1 - According to quantum theory probability exists for sure.

2 - In a coin flip randomness is not the same as the numerous values you can't account for when a person does it.

3 - you must have never been in a casino. I have seen roulette wheels land an amazingly long streak of red or black wins. They post the results for the reason you mention. I have personally lost a chunk of change before because I thought surely it won't be red an 10th time...

They are talking about hypothetical coins.

Bro...

Only a fool would base a statistical experiment on a set of data this limited. Had you flipped the coin only once, and gotten heads, you could say that the coin has a proven 100% chance of flipping heads. You assume that events happens in a pattern that guarantee you get 4 flips of 5 tails in the time it takes to flip 7 heads, the only thing that is safe to assume is that you are 4 times more likely to flip 5 tails them 7 heads.

This so reminds me of playing wow, where people a dumb enough to believe that if creature has 5% chance of dropping the item you need, you have 100% chance of getting the item after killing it 20 times.

The OP claims that the odds of a coin toss are determined by previous coin tosses.

This is easily refuted. By his logic, flipping four coins in sequences has a one in 16 chance of landing all heads. Therefore when you've flipped three heads in a row, only one time in sixteen will that fourth flip yield heads. The other fifteen will result in tails.

As the odds of getting three heads in a row is one in eight, if I did 128 (8*16) coin tosses, I should get only one sequence of four heads. To the random bit generator!

The follow is 64 bytes. That's 128 sets of 4. I will assume each set of 4 is independent of the others in order to keep this simple. I have bolded all the sequences of "1111".



Now the claims of the OP indicate he expects that for every "1111" there should be fifteen "1110". The reality of the sample above shows that they are actually evenly distributed.

Another interesting point: There were two separate runs of seven.

9/10

You got plenty of people to bite.

I didn't make it up. People who hold PHDs come up with this crap not me.

Ph.D*

I believe that PhD is also accepted. In forum speak who knows what is socially acceptable.

Computer science and industrial automation engineer by training, switched a few jobs since I graduated from the university a decade ago, from IT&telecom liason for the Romanian national power company, technical writer and equipment coder for Mitsubishi, then electrical power station design (mostly the automation part) to most recently game designer (the economy part mostly) for a web-based MMO.



The odds of getting 7 heads in a row if you throw it 7 times is 1:128 (0.78125%).
But the odds of throwing at least 7 heads in a row if you throw it 8 times is actually 3:256 (1.171875%), because you have 3 out of 256 possible combinations where at lest successive 7 heads pop up (7h+1t, 1t+7h, 8h).
The odds of throwing at least 7 heads in a row if you throw a coin 9 times is actually 8:512 (1.5625%) because the combos that include 7h or more in a row are 7h+t+t, 7h+t+h, t+7h+t, h+t+7h, t+t+7h, 8h+t, t+8h, and 9h.
The more coins you throw, the number of combinations that include 7 heads in a row divided by the number of total possible combinations of that many throws grows too.

There's nothing there telling you making an insanely improbable throw series is actually impossible, even with a small number of total throws, it only tells you how likely or unlikely it is to get it.
The math only tells you how many throws you need to make to have a good chance of actually getting that many throws in a row.
That chance is not very easy to calculate EXACTLY, however it is much easier to APPROXIMATE.

For instance, take the example of 7 heads across 70 throws.
If we wanted to compute the ACTUAL chance to get 7 heads in a row in 70 throws, we'd have to count all the possible combinations which include at least 7 heads in a 70-throw series and divide that by the total possible combinations (2^70=1,180,591,620,717,411,303,424), but that's pretty damn painful to accurately calculate.
I'm pretty sure there are some decent methods for actually calculating that without actually counting everything, but I'm not really in the mood for exact calculations of that magnitude yet.

For the most conservative inaccurate estimate, we APPROXIMATE that with 10 separate sets of 7 throws (it is an approximation AGAINST the actual chances to throw 7 heads because we discard the possibilities of things like 4h at the end of one set followed by 3h at the start of the very next set, so in reality, the result we'll get is LOWER than the actual chance if we would have considered all the possibilities).
Each of that separate set has a 1:128 chance to get what you want. So there's a 127:128 chance to NOT get what you want. With 10 separate sets, that's a (127/128)^10~=92% chance to NOT get what you want, so a ~8% chance to GET what you want.
As previously mentioned, in reality, the chance to actually get those 7 heads in 70 throws is actually noticeably higher than 8%, because we only considered completely disjointed sets.
Using THIS type of approximation, we would conclude that we'd be having only a ~1.55% chance to get 7 heads in a row out of 14 throws, when we already know that we already get a ~1.56% chance of that happening from as little as 9 throws.

Another inaccurate (and usually more generous than the reality) approximation would be to consider 64 possible sets of 7 heads (starting from position 1-7 to positions 64-70) each with 1:128 chance of happening, leading to a ~60.5% chance to NOT get what you want, and a ~39.5% chance to get at least 7 heads in a row out of just 70 throws. This is obviously incorrect (since we take into account overlapping sets), but it will be used to gather an upper cap for our estimates.
Using THIS type of approximation would lead us to believe that out of 9 throws (3 sets) we would get as much as a ~2.3% chance of 7 heads in a row, when we know in reality it should only be ~1.56%.

So what we can quickly tell from those approximations ?
That that the chance to get 7 heads in a row out of only 70 throws is noticeably higher than 8% but somewhat lower than 39%.
In other words, not extremely likely, but quite plausible.



Yes, that EXACT sequence you flipped has infinitesimal odds of happening (out of all possible combinations of length 70), but it obviously DID happen, because you just got it
By repeating 70 throws, your chance to get whatever sequence is also tiny, but again, it just happened.
Things that seem highly improbable "at second sight" happen every time, it's nothing unusual
The point however is that getting the EXACT SAME sequence both times, now THAT is something that's nearly impossible.

When you take a single sample, that sample can be ANYWHERE in the sample set.
Depending on the type of distribution of the variable being sampled (linear distribution, normal distribution a.k.a. bell curve, or any other type of distribution), that sample has certain chances of being closer to either of the extremes and other chances of being closer to the average.
In time, the more samples you take, the closer the average of the samples will get to the average of all the actual possible values that exist in that particular distribution.

Take for instance a linear distribution between 0 and 1.
Your first sample has an equal chance of being anywhere between 0 and 1, an equal chance for it to be 0.01457 or 0.97175 or even exactly 0.50000.

Now, if your FIRST sample was close to the extreme, like, say, under 0.1 or over 0.9 (total probability of being lower than 0.1 or over 0.9 being only 20%), your next sample has a much higher chance to be closer to the average (80% chance to get something between 0.1 and 0.9).

If your FIRST sample was somewhere closer to the average, like, say, between 0.4 and 0.6 (total probability to be between 0.4 and 0.6 being 20%), the chance of your next sample being farther away from the average is much higher (since the chance of getting something lower than 0.4 or higher than 0.6 is 80%).

It's only when you combine many independent samples that the average of your samples will tend to get closer and closer to 0.5
It won't always get closer to 0.5, but it USUALLY will.

...

Of course, this also applies to a small subset of samples out of a much larger series too.
And it has other applications too.

Like, say, for instance, you hold a lottery, where you have 100 pieces of paper with numbers 1 through 100 written on them.
Now, have 100 people pick the pieces of paper, one for each. Obviously, SOMEBODY will get the pieces marked 91 through 100, and the average value of all pulled lots is 50.5, while the average value for those guys is 95.5.
Now, put all the papers back, kick out all the people who got 90 or lower, and have those "lucky winners" of pieces between 91 and 100 draw one piece of paper each.
There's absolutely no guarantee these "lucky people" actually manage to get high numbers this time, and the expected average value pulled by them is still 50.5, not 95.5 like before. They might get lucky and get the same or a higher average, but it's much more likely they'll get an average much lower than their previous average, and closer to the actual possible average.
Now, pull back those unlucky people who got numbers between 1 and 10 in the first drawing (average of 5.5) and have them draw 10 lots out of all numbers. This time they will almost certainly be far more lucky than before.

There are plenty of other applications of this.