So, there's a Lightning math/cost thread that tries to estimate the cost of a Lightning by working out the fractional number of "Lightnings per 11-pack" and then just multiplying that out. Unfortunately, that's not really how probability works. The correct math makes the situation look either slightly better or much, much worse, depending on how lucky you think you will be.

I'm willing to assume that the percentage chances of a given crystal hatching Lightning are correct; they seem well founded, and they're based in part on the well-studied JP game. The chances of any "normal" Summon being Lightning are therefore 0.005 (0.5%), and the chances of the 11th Summon in an 11-pack being Lightning are 0.025 (2.5%).

No amount of pulls or money guarantees you a Lightning.

To determine the odds of getting a Lightning in N pulls, the easiest method is to determine the odds of getting no Lightnings in N pulls, and then subtracting that from 1:

P(Lightning) = 1 - ((1-0.005)^10*N * (1-0.025)^N))

It is correct that the odds of getting Lightning in your first 11-pack are a little better than 7 percent (or about 1 in 13.7, if you like your probabilities written that way). That doesn't mean that straight multiplication gives you the odds of pulling her in multiple packs.

What does it mean to be "likely" to see Lightning?

Likely means different things to different people. And these are all probabilities. There is no way to guarantee Lightning. To have better than a 50% chance of pulling her ("winning" the flip of a fair coin), you'll need 10 11-packs (P ~= 0.5297). To have better than a 75% of pulling her, you'll need 19 packs (P ~= 0.7615). With 24 packs (P ~= 0.8365), you'll have better than 5/6 odds, but keep in mind that this is the same as rolling a normal 6-sided die; the chances of NOT getting her are the same at this point as rolling a 1 on that die. You can replace that 6-sided die with a 10-sided or 20-sided die if you pull 31 or 40 packs (P ~= 0.9036 and 0.9511, respectively), but if any of you have played tabletop gaming, you're likely quite familiar with those "natural 1s" on a d20 feel like.

So, the question then becomes, what does this cost? You get 18000 Lapis for each $99.99 Vault of Lapis. The 5000 Lapis 11-pack doesn't evenly divide this price, so the cost of chained summons is a step function.

$100 gets you one Vault, and a 20% chance to inspire jealousy in your fellow redditors.

$300 gets you a 50% chance of Lightning. The other thread implies that this is the approximate cost that would make her "likely". That's true, if you think that you're "likely" to win a coin flip.

You need to spend $600 for a 75% chance of Lightning.

$700 gets you better than 5/6 odds (specifically, 84.8% at 25 pulls).

After spending $900, you still have a 1-in-10 chance of being Lightningless.

$1200 makes you 95% likely to have your Lightning waifu. Unless you rolled that natural 1 on your virtual d20, in which case you have some very expensive salt instead.

EDIT: By request, the amount of packs needed to be 99% likely of seeing Lightning is, at least to me, patently absurd. Sixty-three (63) 11-pulls are needed to cross that magical barrier, at the cost of a cool $1900 worth of Lapis. But, hey, there are only 1-in-100 chances that you're still screwed by the RNG, so that's probably totally worth it, right?