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Actually not true. If the website gets a million unique visitors, there is about a 60% chance of someone winning.

Not necessarily. Just like the lottery, it is possible (maybe even likely) that having about a million hits in a day will have periods where nobody wins for a few days, or that a winner might happen two or three times in one day. I’ve forgotten my college statistics, but I believe there are permutation or combination equations that would explain this concept simply in symbolic terms.

Depends.

Do you roll a random number from a generator until it matches the displayed number?

If yes, if you have 1,000,000 visitors per day, the probability of having a winner is around 63 %.

However, if your random number generator guarantees that each player gets a different number then yes, the probability of having a winner approaches 100% as you get closer to the millionth visitor.