Question for you: which of these two images shows dots that are placed at random, and which does not?
The problem with questions like this is that you already know it's the one on the left that's random, and the one on the right isn't, since you know I'm trying to trick you. But what's going on?
Our brains love to find patterns in random noise. Look at the clumping of the dots on the left; surely that's not random? But it is. The distance between dots will average out to some number, but statistically you expect there to be some deviation from that average, so that some dots will be closer together (making clumps) and some farther apart (making voids). That's what's happening on the left.
On the right, the random pattern that was generated was modified so that the dots would not be too close together. If a dot's position was found to be too close to another, its position was redone until it was a minimum distance from all other dots. What's left is a pattern that we think looks more random, but is in fact highly non-random.
A more detailed explanation of these images is at the blog In The Dark, and he uses it to talk about galaxy distributions. However, it also tells us a lot about our brains. We are instinctively lousy at statistics.
Another great example is this one: imagine you flip a coin ten times, and you keep track. Which of these sequences is more likely?
The answer is they are both exactly as likely. You have a 50/50 shot at a heads or tails on each throw, so any 10-throw sequence is just as likely as any other! But we look at the second sequence and see no information in it. We assume it's just random, and therefore more likely than a sequence where we perceive there is information, like five heads in a row followed by five tails. But each is just as likely.
We view the entire Universe through our senses, and the data are processed by our brains. This gloppy computer is highly sophisticated, but also highly unreliable to give us unbiased information. We see patterns where they don't exist, we see cause where they may be none, and we see intent where there may be randomness.
That's why pareidolia -- seeing faces or other familiar objects in random patterns like oil stains, wood grain, and the odd piece of bark of pastry item -- cracks me up. The brain of a human will interpret that pattern into something familiar, and if that person is religious, they see a religious icon. But they don't seem to hang the same connotation on seeing Abe Vigoda in a nebula and Lenin in a shower curtain, or Ben Grimm in a supernova, or any of a hundred examples I can find easily.
You can be fooled. Remember that, always. It pertains to a lot in life, and a lot in the life of an active skeptic. Fooling people is easy. Getting them to see? That's what's hard.