Basics
Independence: why past draws don't matter
A fair lottery draw is an independent event: the numbers that came out last week have no influence on the numbers that come out this week.
What independence means
Independence means the outcome of one draw carries zero information about the next. The probability that 17 is drawn this week is the same whether 17 was drawn last week, a year ago, or never in recorded history.
You can verify this with data: compare how much consecutive draws overlap versus how much randomly paired draws overlap. If they are the same, the draws behave as if independent — which is exactly what every fair lottery shows.
Independence vs patterns
Independence does not mean history looks uniform. Random processes produce streaks and clusters — that is the signature of randomness, not evidence of dependence. See the clustering topic for why.