There is a ~1/3 chance of awakening the featured mythic in the anniversary eggs with 800 gems (I have not done the calculations for other amounts but the odds are super high regardless). No other eggs are worth spending on unless you have paid gems or several thousand gems saved up. I see people spending their gems on the current banners but they are being baited into wasting their gems!
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Here is the math I did. I used a binomial calculator for a lot of it.
0.44280432263 * 0.5625 + 0.19067830777 * 0.375 + 0.0649328394 * 0.0625 = 0.32464009935 or 32.46%
(2 or more copies without guarantee * 2 copies from guarantee + 3 or more copies without guarantee * 1 copy from guarantee + 4 or more copies without guarantee * 0 copies from guarantee)
This assumes there is a 0.75% chance of getting the featured mythic. I’m not sure if the odds will be the same because this is an anniversary egg and the egg pool will be rotating at the same time