The "margin of error" you read under polls hides a bigger worry: in a close race, a single honest poll can point at the loser. Guess how often, then run the polls yourself.
A candidate truly has 52% of voters, the rival 48%. A pollster asks 1,000 randomly chosen voters. Out of 100 such polls, how many show the rival ahead?
Each dot is one simulated poll. Red dots put the true leader below 50%, so the poll calls the wrong winner.
Chance that one poll shows the wrong leader. Tap a cell to load it into the lab above. Greener means safer. Note the pattern: a thinner lead needs far more than a few times more people.
Quadruple the sample and the sampling error only halves. At 52 to 48, going from 1,000 to 4,000 people drops the wrong-call chance from about 10% to about 0.6%. At 51 to 49 it takes about 10,000 to get under 3%.
Everything here is computed, not sourced: no outside data. Each poll draws n independent yes/no answers with the stated true share. The wrong-leader chance is the exact binomial probability that fewer than half of the sample pick the true leader (a tie counts as half). The margin of error is 1.96 times the square root of p(1-p)/n. Real polls also face nonresponse, weighting and question wording, so their true error is usually larger than this ideal.