Odds of 200 Orbs and no 5-Star?

I spent 100 this morning, got nothing. Did full summons each time.

So:
(1-.08)^5*(1-.085)^5*(1-.09)^5*(1-.095)^5*(1-.1)^5 = 0.0946

Or 9.5% chance of NOT pulling a 5* in 100 orbs while doing full summons.

So for 200, it's

(1-.08)^5*(1-.085)^5*(1-.09)^5*(1-.095)^5*(1-.1)^5*(1-.105)^5*(1-.11)^5*(1-.115)^5*(1-.12)^5*(1-.125)^5 = 0.00445596941

or 0.4% chance of not pulling a 5* within 200 orbs. You would be at 13% pity rate. It has happened, I know I have gotten to 12.5% on one of these banners before. If you are sniping, you have a slightly lower chance of pulling a 5* per orb on average because you might be pulling up to 9 summon orbs at 9% before it goes up to 9.5%.

I believe it's as simple as putting in 92% (Odds of not getting a 5*) and using to the power to 5 and basic multiplication with every percent increase. The odds of getting a 5* from a full summoning round on a legendary banner with 8% is around 44%. That's high, but only because you're not going for anything specific. If you were aiming for Green, the most uncommon colour(For the highest chance at a specific colour), that chance drops by quite a bit.

I spent 260 orbs on the previous legendary banner without a 5*, then got a B!ke (wanted Hector obviously). I was sniping, though.

My advice, don't even ask how small the chances are. The odds will just make you feel worse. Instead, rejoice over the fact that you are playing the odds in the smartest way possible for you and the goals you are trying to achieve. And if that's not the case, rethink your approach to the game. Both of these things are what I do, and it helps you making smarter choices and getting over a bad summoning streak. I'm sitting at 285 orbs and counting, because this new banner is just terribad.

Karma will come back around for you. For instance, after my insanely unlucky Legendary Hero banner summoning, I free summoned a (-spd+res) Sonya, Amelia and a (-atk+spd) Swordhardt. Pretty decent for a F2P player, even with the god awful IV's.

Honestly, does it really matter what the odds are? It's highly unlikely, but not impossible. I've spent around 150 orbs (at least) several times and not gotten a single 5-star. The odds of that happening are low, but that doesn't mean it didn't happen. I think people put too much faith in odds. Things either happen or they don't, especially when it comes to games of pure chance. Knowing you had low odds of not pulling anything despite significant investment won't make you feel any better if it happens to you. (Trust me; speaking from experience.) People want to feel like they have some control over the situation and that RNG isn't straight-up gambling, and that gambling isn't pure chance, but that's all just a way that we fool ourselves into making bad decisions. Don't worry about the "odds". Just do it if you want whatever you're going for badly enough to risk wasting 200 orbs without getting it, because it might happen. OTOH, you miss out on 100% of the things you don't even try for.

I was at 5.25% on the halloween banner. I was pulling everything except blue and I finally got H Kagero out of it. I am still trying and I just got a Faye which I will feed to a merged Rebecca.