You have to be very specific about "probability that they occurred by chance." If p = 0.05, you can't say "there's a 95% chance this result is real and a 5% cha

You have to be very specific about "probability that they occurred by chance." If p = 0.05, you can't say "there's a 95% chance this result is real and a 5% chance it's just a fluke." You can say "if chance is the only thing operating, we'd see a result like this only 5% of the time." In conditional probability notation, it's the difference between P(result | it's just chance) and P(it's just chance | result).