Because the third estate was 90% of all citizens, so it would definitely win if the estates voted per person. The first and second estate were against, and the nobles knew that if they let each estate vote only once, they would get their way. It was 2 against 1.
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The Third Estate, made up of about 98% of France's population, which included the bourgeoisie, peasants, and laborers. Vote by Estate meant that each Estate was entitled to one vote per Estate, so the First and Second Estates (made up of about 2% of France's population, the aristocracy and the clergy) could vote together and successfully vote against 98% of France. Voting by head meant that the Estates would vote based on population, meaning that the Third Estate would have much more votes than the First and Second Estates.
The system was that each estate took its own poll on a one-man-one-vote basis; the majority in each estate decided how that Estate should vote; and the final voting was by Estates. On tax questions, the Third Estate was bound to be outnumbered two to one, despiteits memebrs being far more numerous than those of the other two. Naturally, threfore, they wanted a single one-man-one-vote arrangement.
The members of the Third Estate were arguing about the process for the Estates-General. The king's ministers had intended that the three estates meet and vote separately, with each estate having one vote. Instead, the Third Estate insisted that the three estates meet as one body, and that voting be by head so that each person have one vote (the Third Estate had as many members as the other two estates combined). When the deadlock over procedure could not be resolved, the members of the Third Estate declared themselves to be the only legitimate representative body, and renamed themselves the "National Assembly". They then vowed never to disband until they had written a constitution for France. They did this on 20 June 1789. The Third Estate invited the members of the other two estates to join them as part of the National Assembly, which they eventually did.
The Third Estate had as many members as the other two put together; but it was decreed that each Estate had ONE vote. That was why the Third Estate declared itself to be the National Assembly, and that declaration (the Tennis Court Oath affirmed it) was the start of the Revolution.
The Clergy and the Nobles had a symbiotic relationship and since each Estate had a single vote there was no way that the vast majority of the Third Estate could ever pass legislation that dealt with equality, Democracy or changes to the taxation system.