If I toss a coin. What it the chance that I get 10 consecutive heads (or tail)? Is it (1/2)^10=0.00097? Many people think, if you see 9 consecutive heads, than the chance seeing next one is head is extremely small. And you should bet tail. But some people said, the chance is still 1/2 because they are independent. I am getting confused. Some math wiz bother to explain?

There is a difference between the probability of 10 in a row and any individual throw. In fact, you used the fact that the probability of a head on any individual throw is .5 to derive the probability of 10 in a row. So the odds are 50/50 for number 10. Frankly, I have never really understood this and still don't, even though I know the above is correct. How can you have different probablities for the same event for the next throw? In the markets, there is one big difference. Unlike a coin, the market knows it has been up "n" days in a row. So I think the odds for succeeding days decrease.

I believe the key here is 'mutually exclusive'. Despite the appearance that tails would almost be a guarantee I believe the probabilities are still 50/50. Each flip of the coin is mutually exclusive and there is still 2 possible outcomes, that probability does not change. But that being said I know there are more complicated ways to analyze this and I am neither a math wiz nor a stats wiz. Interesting story regarding this problem. I have a friend who used to own a gas station and every time I went in to fill up my car (at least once a week) we would flip for 20 bucks. I always took heads he always took tails. We used to keep track of it on a big wall calender that he had hanging up. Guess what the outcome was at the end of the year. For two years it was almost dead even, one year I was up like $80 the other he beat me by $40. Interestingly both of us had our runs but it would always even out. Oh yeah the 40 bucks that I netted out.... spent it on beer that we both drank together so it did even out exactly! MACD

Because you are looking at the same event in different contexts. The probability of the coin landing on heads ten times in a row is relatively small. The probability of this happening is made up of a series of events occurring. That is, the probability of the coin landing on heads, once, twice, thrice, four times, and so on. The key is, you have to bring all these events together and look at them as a series, rather than each being a separate throw. So, that is why the probability of flipping heads ten times in a row is so low. The other question is what is the probability of flipping heads on the tenth try, after all the previous ones have been heads? The answer is 1/2, or 50%. Assuming the coin is fair, there are only two outcomes, heads being one, tails, the other. No matter what has happened during the series, the chance of the tenth flip being a head is 50%. Here we are not looking at the flip in a series, but as an independent event. The key phrase in this instance is "tenth try". We are looking at one event, not a series. Also, To MACD - The outcomes are both mutually exclusive and independent of previous trials. Mutually exclusive, because you can have either heads or tails, but not both (assuming the coin cannot land on its side). Independent of previous trials, because the coin does not have a "memory", means the outcomes prior to the current trial have no influence on the current trial (thereby leading to the inference of a 50/50 chance of heads or tails on any given flip). Additionally, that most people would "bet tails" is a gambler's fallacy. That is, since it has been heads nine times in a row, the tenth flip "has got to come up tails". The chance of heads on any one throw is no different than any other. I think this is correct. And, I hope this helps.

Here's a neat idea to help you understand things like this. Take a big jar of pennies, and throw them up in the air. Remove all the tails, and throw the rest up in the air. Repeat until you have only 2 or three coins left. Take one of those pennies out and toss it in the air. Do you expect it to land heads again? After all, it has only landed heads for many throws so far. . .

just some simple stochastics... each unique toss has a probability of 50% for each side. But as soon as you see the whole thing in a series you have to multiply all the events: 0,5^10 = 0,098% but the probability for the 11th toss to land on e.g. head is again 50% BUT the chance to get a SERIES of 11 consecutive heads is only 0,049%