stupid question

[slim]

I know that each coin toss of a fair coin is an independent event no matter what happened on the previous tosses. So if you tossed a coin 1 million times, there should be close to 50-50 head/tails. But what if u tossed the coin 250,000 times and by some extreme luck factor, u got 75-80 % heads and u could wager on the next 750K tosses. Isn't it more likely to be tails in order for the distribution to normalize?

[Kerth]

What exactly would cause the coin to land on tails more often during the next 750k tosses?

[drbst]

No. The weak law of large number only states that the frequencies approach the expected values.

Example: 100 tosses, 51 heads, 49 tails, frequency 51% vs. 49%, absolute difference: 2

1000 tosses, 510 heads, 490 tails, frequency 51% vs. 49%, absolute difference: 20

In fact, there is another result from random walk theory that states that if you play a symmetric game, the guy who has the lead in the beginning will most likely be the winner even after a long period.

[ThinkQuick]

[ QUOTE ]
I know that each coin toss of a fair coin is an independent event no matter what happened on the previous tosses. So if you tossed a coin 1 million times, there should be close to 50-50 head/tails. But what if u tossed the coin 250,000 times and by some extreme luck factor, u got 75-80 % heads and u could wager on the next 750K tosses. Isn't it more likely to be tails in order for the distribution to normalize?

[/ QUOTE ]

You're right, it took some crazy luck to get you to that point, but you got there. from there, the most likely scenario is that you will get 50/50 from now on.. you have no advantage over the next 750k tosses. This has been posted a few times before..

i.e. you got 200K heads in your first 250K flips. I am assuming that this extremely rare event is due to luck alone as you said, although if this really happened, you can assume the coin is weighted.

So you have 750K flips left, and the results of those should approach 50/50, so you'll most likely get around 375K heads.
Add this to the 200K you already have, and you can see that if your first 250K flips are as messed up as they are, the final results are no longer most likely 500K:500:, but now 575K:425K.

Note that this 75K swing after a million flips is much less of a deviation percentage wise than it was after only 200K flips. This is exactly what the law of large numbers predicts.. This is the reason that there "should be close to 50/50 heads/tails". If you flipped it 10million times, then even the craziest luck during the first 250K hands would have hardly an effect on the final ratio after all flips.

[Precision1C]

If I was getting even money on my bet I would bet on tails since if the coin is fair it doesn't matter which I bet on and since the results were so skewed toward tails for the first 750k it looks like the flip method or the coin may be flawed.

[bav]

[ QUOTE ]
I know that each coin toss of a fair coin is an independent event no matter what happened on the previous tosses. So if you tossed a coin 1 million times, there should be close to 50-50 head/tails. But what if u tossed the coin 250,000 times and by some extreme luck factor, u got 75-80 % heads and u could wager on the next 750K tosses. Isn't it more likely to be tails in order for the distribution to normalize?

[/ QUOTE ]

In this case I'd start wagering on heads, myself. It's pretty unlikely it's actually a fair coin. [img]/images/graemlins/smile.gif[/img]

You're saying the words here, "know that each coin toss of a fair coin is an independent event no matter what happened on the previous tosses" but then go right on and say the opposite in the next sentences. You're saying it, but not believing it. "independent events". What happens next doesn't depend on what happened prior.