The way I've argued about probability in the past is that 10 heads in a row is just as likely as any other specific outcome, for instance HTHTHTHTHT or HHHHHTTT

The way I've argued about probability in the past is that 10 heads in a row is just as likely as any other specific outcome, for instance HTHTHTHTHT or HHHHHTTTTT. The coin flip events themselves are not related to each other, as the outcome of one flip doesn't affect the next. So it doesn't really make any sense to look for trends. I was actually talking about colors in roulette, but the coin flip is a pretty close example. This seems extremely logical when you think about it, but a lot of people don't look at it this way. They always say something like, "Well, black has been coming up a lot lately, so it's definitely going to be red soon."