Bernoulli Trials: Law of Large Numbers vs Gambler's Fallacy, the N paradox

I have asked this question before but I think it wasn't clear what I implied with my succinct question, so I will be a bit more verbose this time.

Lets set the following example: Bernoulli trials, K=17 p=0.525 N=20,000

The probability of a streak of at least 17 consecutive successes in 20,000 trials is 15.3% The same but with N ten times larger, the probability of a streak of at least 17 consecutive successes in 200,000 trials is 81.01%

So my question is the following: is the probability of getting 17 consecutive successes still 81.01% if I run 10 independent trials of N 20,000?

If the law of large numbers are correct, nothing should change since N is simply incrementing, 20,000 today and 20,000 tomorrow is the same as running 40,000 straight, right? So what happens when I run 20,000 on the tenth day? Does that last 20,000 really have 81% of winning 17 streaks just because it is totaling 200,000? That definitely sounds like the Gambler's fallacy. If we consider that each trial is random and independent, 20,000 should always represent 15.3% regardless of how many times we run it...

It should be indistinct to be tossing the coin 200,000 nonstop and tossing 10 times groups of 20,000. How on Earth would pausing and resuming tosses change anything? Right? On the other hand each group of 20,000 tosses are independent and random so there is no way its probability of getting 17 streaks should increase.

So what is the right answer?

• This is not clear. Suppose My first block of $20000$ ends with $10\;H's$ in a row, and the next starts with $7\;H's$ in a row. Does that count? If it does, then you are back in the $200000$ case. If it doesn't, you aren't.
– lulu
Jul 17, 2016 at 12:41
• To make the contrast more plain: suppose you split your $200000$ trials into groups of ten. Then there is $0$ probability of getting seventeen $H's$ in a single block.
– lulu
Jul 17, 2016 at 12:42
• I guess that makes sense. I guess I wasn't considering the splitting of a streak between the groups. So there are 10 opportunities of breaking the streaks. But why are you saying there is 0 probability of getting seventeen H's in a single block? Isn't it still 15.3% according to the Bernoulli trials? Jul 17, 2016 at 12:47
• If I have blocks of length ten, then it is impossible to get $17\;H's$ in a row within a single block, obviously.
– lulu
Jul 17, 2016 at 12:50
• My extreme example just amplifies your observation. Using groups of length ten, it is impossible to get the desired streak within a single block, yet of course there is a very high probability that we get the streak if we ignore the separation into blocks. All that means is that, as the block size decreases, the probability that a favorable streak spans multiple blocks increases.
– lulu
Jul 17, 2016 at 12:53

not quite the same, you could get 7 at the end of one trial and 10 at the beginning of the other, but that is a minor error. What you are essentially doing is running 1 bernoulli trial with 20k paths and $p = 15.3\%$, and then repeating it 10 times, so not getting any success has chance of $$(1-p)^{10}$$ which is indeed a reasonably small number...