Ouyang Wulong made a mistake when he said that if you flip a coin ten times and come up with all heads, for the next set the coin is more likely to come up tails to "balance out" the first set.

The probability of flipping a fair coin ten times and coming up with all heads is 1 out of 1024. It doesn't matter if this is your first set of 10 coin flips, your second, or your third.

I think the point he was making is that in the short run, you can get weird results (like flipping a coin 10 times and getting heads all ten times) but in the long run random phenomena take on a predictable pattern (if you flipped a coin a million times, 50% the time you are going to get heads, 50% the time you a...

Ouyang Wulong made a mistake when he said that if you flip a coin ten times and come up with all heads, for the next set the coin is more likely to come up tails to "balance out" the first set.

The probability of flipping a fair coin ten times and coming up with all heads is 1 out of 1024. It doesn't matter if this is your first set of 10 coin flips, your second, or your third.

I think the point he was making is that in the short run, you can get weird results (like flipping a coin 10 times and getting heads all ten times) but in the long run random phenomena take on a predictable pattern (if you flipped a coin a million times, 50% the time you are going to get heads, 50% the time you are going to get tails).

It is probably human nature to perceive predictable random patterns as one trial having an effect on the subsequent trials. However, if the phenomenon is truly random, the outcomes of the trials are independent of each other. The results of flipping a coin 10 times have no effect on the results of the next set of coin flips. Thinking otherwise has been the downfall of many gamblers.

In response to your concerns about the independence of probability, you aren't considering that the entire system also needs to conform to the 50% probability as well. The farther a system strays from it's natural probability the less likely it is to remain that way. More important that your individual coin flipping approaching 50%, is that all coin flipping ever also approaches 50%.

The odd results are actually, more often that individual coins will flip more heads than they should, while other individual coins flip more tails than the should, although the system maintains its 50% equilibrium. There are some cool java applets out there that illustrate this.

The reason for the gambler's fa...

In response to your concerns about the independence of probability, you aren't considering that the entire system also needs to conform to the 50% probability as well. The farther a system strays from it's natural probability the less likely it is to remain that way. More important that your individual coin flipping approaching 50%, is that all coin flipping ever also approaches 50%.

The odd results are actually, more often that individual coins will flip more heads than they should, while other individual coins flip more tails than the should, although the system maintains its 50% equilibrium. There are some cool java applets out there that illustrate this.

The reason for the gambler's fallacy is that the gambler is not capable of considering the overall system of which he may be a part.

Actually, an even better way to put it, tying it in to arifsali's clip, good or bad luck could be viewed as a product of continuity in our individual perspective. A coin's behavior in a group is observable to us (although it isn't easy to keep track of the individual coins if you try this...) but only as part of the whole. The coin does not have its own perspective, and does not evaluate its own flips.

Thus, the system can be analyzed in terms of unique tests, or events. However, when it comes to people we are inordinately cognizant of our own happenings and fail to consider them as part of a larger system. Thus, sometimes strange results for any one individual can seem like good or bad luc...

Actually, an even better way to put it, tying it in to arifsali's clip, good or bad luck could be viewed as a product of continuity in our individual perspective. A coin's behavior in a group is observable to us (although it isn't easy to keep track of the individual coins if you try this...) but only as part of the whole. The coin does not have its own perspective, and does not evaluate its own flips.

Thus, the system can be analyzed in terms of unique tests, or events. However, when it comes to people we are inordinately cognizant of our own happenings and fail to consider them as part of a larger system. Thus, sometimes strange results for any one individual can seem like good or bad luck, but they are really part of the standard probability in the overall system.

In response to your concerns about the independence of probability, you aren't considering that the entire system also needs to conform to the 50% probability as well. The farther a system strays from it's natural probability the less likely it is to remain that way.

Look at the formula in your clip--it doesn't say anything about balancing out previous results, or correcting a system that has "strayed" from its natural probability.

The probability of flipping a coin ten times and getting 10 heads is .5^10 ("p^k"; the other terms simplify to 1 because both n and k are 10). If you do a set of 10 coin flips and get 10 heads, your chances of getting 10 heads on the next set ar...

In response to your concerns about the independence of probability, you aren't considering that the entire system also needs to conform to the 50% probability as well. The farther a system strays from it's natural probability the less likely it is to remain that way.

Look at the formula in your clip--it doesn't say anything about balancing out previous results, or correcting a system that has "strayed" from its natural probability.

The probability of flipping a coin ten times and getting 10 heads is .5^10 ("p^k"; the other terms simplify to 1 because both n and k are 10). If you do a set of 10 coin flips and get 10 heads, your chances of getting 10 heads on the next set are exactly the same (1 out of 1024). Your chances of getting 10 tails are also the same (1 out of 1024). There is no adjustment to the formula for what happened in the prior set.

The odd results are actually, more often that individual coins will flip more heads than they should, while other individual coins flip more tails than the should, although the system maintains its 50% equilibrium.

Yes, but this isn't because of some "need" of the system to restore balance or achieve equilibrium. If you flip a coin a million times, you are going to eventually see 10 consecutive heads somewhere among those million flips. You are also going to see 10 consecutive tails somewhere else among those million flips . Not because of some need to cancel weird results out, but because 10 consecutive heads (or 10 consecutive tails) are just common enough to show up in a million coin flips.