First of all, it’s possible to imagine a mutation that makes you have 10x the number of children of a regular person. This gives us s=9. Is the probability of fixation then… 18?
Good spot. The P(fixation) = 2s formula is actually an approximation that holds only when s is small but positive and the population size is very large. A more precise formula (here’s a publicly-accessible paper deriving it) is P(fixation) = (1 - exp(-2s)) / (1 - exp(-4N**s)), where N is the population size. When s is small one can approximate the numerator by 2s and when s is positive and N is infinite the denominator becomes 1 — so when all of these conditions hold the formula reduces to 2s. In your example s is large and the P(fixation) = 2s approximation is no longer a good one.
Good spot. The P(fixation) = 2s formula is actually an approximation that holds only when s is small but positive and the population size is very large. A more precise formula (here’s a publicly-accessible paper deriving it) is P(fixation) = (1 - exp(-2s)) / (1 - exp(-4N**s)), where N is the population size. When s is small one can approximate the numerator by 2s and when s is positive and N is infinite the denominator becomes 1 — so when all of these conditions hold the formula reduces to 2s. In your example s is large and the P(fixation) = 2s approximation is no longer a good one.