Randomness Is Random

From The Better Angels of Our Nature - this is a bit off topic from the theme of the book, but I always find this type of thing (examples where our intuitions are wrong) really interesting:

Suppose you live in a place that has a constant chance of being struck by lightning at any time throughout the year. Suppose that the strikes are random: every day the chance of a strike is the same, and the rate works out to one strike a month. Your house is hit by lightning today, Monday. What is the most likely day for the next bolt to strike your house?

The answer is "tomorrow," Tuesday. That probability, to be sure, is not very high; let's approximate it at 0.03 (about once a month). Now think about the chance that the next strike will be the day after tomorrow, Wednesday. For that to happen, two things have to take place. First lightning has to strike on Wednesday, a probability of 0.03. Second, lightning can't have struck on Tuesday, or else Tuesday would have been the day of the next strike, not Wednesday. To calculate that probability, you have to multiply the chance that lightning will not strike on Tuesday (0.97, or 1 minus 0.03) by the chance that lightning will strike on Wednesday (0.03), which is 0.0291, a bit lower than Tuesday's chances. What about Thursday? For that to be the day, lightning can't have struck on Tuesday (0.97) or on Wednesday either (0.97 again) but it must strike on Thursday, so the chances are 0.97 x 0.97 x 0.03, which is 0.0282... With each day, the odds go down... because for a given day to be the next day that lightning strikes, all the previous days have to have been strike-free, and the more of these days there are, the lower the chances are that the streak will continue...

... events that occur at random will seem to come in clusters, because it would take a nonrandom process to space them out.

The human mind has great difficulty appreciating this law of probability. When I was a graduate student, I worked in an auditory perception lab. In one experiment listeners had to press a key as quickly as possible every time they heard a beep. The beeps were timed at random... The listeners, graduate students themselves, knew this, but as soon as the experiment began they would run out of the booth and say, "Your random event generator is broken. The beeps are coming in bursts." ... that's what randomness sounds like.