First off I would like to say that I am a probability noob so I apologize if this question is dumb.

Experiment 1: You are doing an experiment where you flip a coin a bunch of times and record the results by writing tally marks under a "Heads" column or a "Tails" column. The more you flip the coin, the closer your results will be to being exactly X / X. Where X is your number of flips divided by 2. (Note that "/" here is not being used to denote division, it's being used to separate Heads and Tails results. An experiment that consisted of 4 Heads and 9 Tails outcomes would be denoted as 4 / 9.)

Experiment 2: In this experiment you will do everything exactly the same as in experiment 1, except for one thing. You decide to mark 1000 tallies in the "Tails" column before you begin the coin flipping. You now start to flip the coin and record results. This time the results will not tend toward X / X, they will tend toward X / X + 1000.

Experiment 3: This experiment is the exact same as experiment 1. This time by random chance your first 1000 flips are Tails. You are now in the exact same situation as you were in in experiment 2 (An empty "Heads" column and a "Tails" column with 1000 tallies), but this time the results will tend towards X / X. How is this possible?

  • $\begingroup$ Even if by coincidence we get tails on the first thousand tosses, the fraction of tosses that are heads will still approach $1/2$ in the long run. (For example, after a trillion tosses, whatever happened in the first 1000 tosses will be a tiny blip in the total tally.) $\endgroup$ – littleO Apr 7 at 8:57

Nothing will tend to $X/X$ or to $X/(X+1000)$

Think this way... What is X there (in all three experiments)?
Well, it's a free variable so nothing can tend to it, because X keeps changing.

The truth is that yes, in 3) after the first 1000 flips, you're in the same situation as you were in 2). So the number of tails will always be appox. with 1000 bigger than the number of heads.

Now... 1000 is big difference if you made 2000 or 3000 or a few thousand flips in TOTAL. But it's nothing when TOTAL goes to infinity.

So if you make 1 million or 1 billion flips in total, or even more flips, eventually it will stop to matter that you started with 1000 tails, the numbers of heads and tails (even including the results of the first 1000 flips) will be appox. the same (approx. in terms of the TOTAL number of flips).

That's because... in formal terms:

$$ \lim\limits_{X\to\infty} (X/X) = \lim\limits_{X\to\infty} X/(X+1000) = 1 $$

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