The Probability of Informational Cascades

If I'm the one millionth person in line, I've collected up a million signals about whether to buy an iPhone. And if different people are receiving signals that have at least some degree of independence so that the different signals are incrementally informative, then a basic theorem of statistics and about probability updating says that you're going to be able to draw conclusions that are correct almost for sure. In stark contrast, after say three people have adopted the iPhone, I might fall into a cascade. There's almost a 50% chance that that cascade is wrong.