contradiction to law of large numbers? please explain

[csuf_gambl0or]

ok so my (idiot)friend was trying to tell me about his super blackjack system that he thought of. It was basically a martingale system. I was trying to explain to him why its a stupid system and it doesn't work.

He kept saying that if you lose like 7 hands in a row, the chances of you losing the next hand go down and the chances of you winning go up. He compared it to the use of a coinflip.

I kept trying to tell him that each hand, or each coin flip is an independent event. The last hand/flip has no impact on the next one. If the last coinflip was heads, the next flip has the exact same chance of landing on heads as the last flip, they are independent events.

But here in lies the problem. The law of large numbers states that over time, the events should even out. This means that if a fair coin is 50/50 to land on heads or tails then if the coin is fliped say 40,000 times, it will approximately land 20,000 on heads and 20,000 on tails. So if a coin lands on heads 10 times in a row, then theoritically the chance of the next flip being heads GOES DOWN right?!?! But how can this be? Doesn't the theory of independent events and the law of large numbers contradict each other? Someone please explain and clarify for me.

[Ra_]

they dont contridict, if you flip a coin 5 times and it lands on heads, and then you flip it a million times, on average you will get 500,005 heads and 500,000 tails. so as the more flips you do the closer you get to 50% heads, but they are still indepentent.

[AaronBrown]

The law of large numbers refers to the proportion of heads, not the number.

If you flip a coin and the first ten times it comes up heads, you have 100% heads. The law of large numbers says by 40,000 flips, that proportion will likely decline.

You also have ten more heads than tails. The independence of coin flips tells you that you expect to have ten more heads than tails after 40,000 flips. So you expect 19,995 of the next 39,990 flips to be heads, and 19,995 to be tails.

These statements do not contradict each other. If you get the most likely outcome, exactly 19,995 heads out of the remaining 39,990 flips, you will have 20,005 heads and 19,995 tails, for 50.0125% heads. The number of excess heads (ten) hasn't changed, but the proportion of heads has gone down from 100% to 50.0125%. Most important, you did not expect more or fewer than half the 39,990 flips to be heads, the first ten flips did not affect that expectation.

Of course, even though 19,995 heads is the most likely outcome, it's not very likely. You could get 19,900 or 20,100 or any other number of heads. All you know is deviations of more than a few hundred from 20,000 should be extremely rare. So you could end up with between 49% and 51% heads, but it's very unlikely the proportion will be outside that range.