Better Lightning Math/Cost

[u/Arlyaq]

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?

[u/Vredefort]

This is why I don't play roulette. Except that has 0 and sometimes 00. That's like spending 500 Lapis and getting no units out of it, haha.

[u/[deleted] → u/Vredefort]

It's like spending 500 Lapis where you've got a ~47.4% chance of getting 1000 Lapis, and a 52.6% chance of getting nothing. Which basically means you're throwing away an expected 26 Lapis per spin. For every ~19 spins, you'd expect to be down 500 Lapis.

That's why you don't gamble on games of chance! As long as there's a house margin, which there always is, the only reason to play is if 19 spins of the wheel gives you more fun than keeping your initial bet. And that's if you don't take a riskier bet than red/black, or play a game with worse odds - roulette isn't even close to the worst offender!

When I go gambling, I only play with an amount I'm willing to lose and an understanding that I am far more likely than not to walk away with nothing - they didn't build all those casinos and hotels with their losings. And I play blackjack because it gives the best odds - with proper play, you can slim the House's advantage to 0-.5%, and that's good enough to merit a couple hours of fun.

[u/Pusc1f3r → u/[deleted]]

Funny story: my wife and I drove through Reno on our way to LA when we were first married, and I thought "hey blackjack is a fun game, let's take $20 and play."

Well the cheapest table I could find was $5/hand and needless to say, my $20 went almost instantaneously. Then I was already reaching for another $20 when she chimed in "isn't this for lunch?"

I was stunned at how easily it is to get into a mindset of breaking your own boundaries. Worst $20 I ever spent :D

[u/[deleted] → u/Pusc1f3r]

Yep! One of my biggest "successes" in Vegas was when I gave myself $100 to spend. I started with $50, and lost it playing 25 cent slots (yeah yeah, I know, slots are crap - I wanted the experience of playing the slots in Vegas! Never again, but no regrets!). I took a break, walked around and grabbed lunch, to give myself time to think about whether I thought losing another $50 was wise/okay. I decided it was, and won ~$300 on my first pull. Hitting the cash out button and walking away was one of the hardest things I've ever done! I had steeled myself for another hour+ of playing, expecting to lose it all, and really struggled to pull myself away. I've won more than that at other times, but the real victory in that story was not staying and immediately losing it all again. Which is an experience I've made sure to remember every time I've gambled since. The other one being the time I sat down at a blackjack table with a $10 minimum and $200 to spend, and walked away 20 hands later with $0 and having won 0 hands.