Why does a random number change your next guess?

A random number changes your guess because your mind uses whatever number it saw last as a starting point, then doesn't move far enough away from it. Psychologists call this the anchoring effect: a reference point, even a completely irrelevant one, drags your estimate towards itself.

In a famous experiment by Amos Tversky and Daniel Kahneman, people watched a roulette wheel that was rigged to stop on either 10 or 65. Then they were asked what percentage of United Nations members are African countries. Those who saw 10 guessed 25 percent on average; those who saw 65 guessed 45 percent. The wheel obviously knew nothing about Africa, yet it moved the answers by twenty points.

The most common explanation is called anchoring and adjusting. When you're unsure, you grab a starting value and nudge it up or down until it feels reasonable. The trouble is that you stop nudging too early, so your answer stays close to wherever you started. Students asked whether Gandhi died before or after age 9 guessed, on average, that he died at 50. Students asked about age 140 guessed 67. Nobody believed either anchor, and it still worked.

That's why the first number in a negotiation matters so much: studies find initial offers shape the outcome more than the counteroffers that follow. Even being warned doesn't fully protect you. The most reliable defence found so far is to deliberately consider the opposite: ask yourself why the true answer might be far from the number you were just shown.