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You must distinguish them because expected value measures the average financial gain or loss relative to the initial investment. The prize paid is the gross revenue, but the net profit accounts for the cost of the ticket.

Conditions: The lottery ticket has a non-zero cost.; The calculation aims to determine the player's expected financial outcome (profit/loss).

You must distinguish them because expected value measures the average financial gain or loss relative to the initial investment. The prize paid is the gross revenue, but the net profit accounts for the cost of the ticket.

Conditions: The lottery ticket has a non-zero cost.; The calculation aims to determine the player's expected financial outcome (profit/loss).