first prize in a private lottery
Refers specifically to the top prize in a private lottery (not a public one). The term is somewhat specialized and may not be widely used in everyday conversation.
I won the first prize in the shopping district's lottery.
本 (main/real) + 籤 (lottery/slip). The exact historical derivation is uncertain, but the compound literally suggests the 'real' or 'main' winning slip in a lottery.