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If there are only 330 permits don't the odds depend entirely on how many people sign up?

Sure, but they could display odds from past lotteries or provide current odds at time of purchase. You really have no clue what you're getting when you give them your $6-$15.

Perhaps the above comment means that the secondary factors affect the denominator - i.e more people go hiking on Saturday, so entering the Saturday lottery is worse than entering the Wednesday one

Exactly, and it impacts the denominator by a great margin.

The value of the lottery ticket to me is a function of the cost, odds, timing and interest I have in the destination.

If the odds of one trailhead are 1000x lower than another one that is comparable across the other variables then I'm needlessly overpaying, and Booze Allen pockets that inefficiency.

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