The math...
http://wiki.spiralknights.com/Equinox_Prize_Box_Promotion_March_2014
There's a 1% chance of getting a rare Equinox item.
OK, so...
.99^x=.5
xlog.99=log.5
x=log.5/log.99
x ~= 69
69*$10=$690
OK, so...
...you could buy...
...an XBox360, PS3, yearlong subscriptions for both, an extra controller for both, and a handful of games...
...or...
...you could buy a coinflip on a shiny costume in this game.
The choice is yours.
Uh, what? So, you're calculating how many times that you don't get something that you have a 1% probability of getting such that it's the same as the probability of getting something you have a 50% chance of getting why?
Edit: Ok I've wrapped my head around why you're doing it, but still, it's not the most useful metric for thinking about this situation...
Surely if its $10 for 1 box and there is a 1% chance of getting something rare, then the chance is 1 in every 100 boxes that you'll get a rare so assuming you do get a rare after 100 boxes, its $1000 for a rare :u
It's not that you pay $690 dollars to get a definite chance of getting a rare costume from the equinox promotion. It's possible to be unlucky and not get the rare costume after paying $690.
You've declined to mention the 315000 CE that would accompany that purchase.
I'm sure you could buy quite a few of the 1% items with that.
Or 22500 Radiants.
You could single handedly end the Radiant drought!
Or you could just say "Screw this, Spiral Knights is good but it's not 100 dollars good".
My bad guys.
The math should also include the probability of getting multiple items as well as the energy you actually get...
...but considering that the energy market has plummeted 1,100 crowns yesterday from 7,100 to 6,000 per 100 energy, I'm just not motivated enough to bother. Apparently, people are still buying tons of energy to get the costume pieces anyway.
69 boxes =
Your math is off, you've done "0.99^69" which is technically implying that you roll 69 times and the 69th time is the determining factor, it either rolls successfully or not. What you should be doing is calculating chance on a combination basis, the chances that you'll come out with either a success or a fail per roll and different combinations of successes and fails, aka "chance of winning just one = chance of failure^fails * chance of success^successes * combinations"
50.0% chance of obtaining zero 1% items: 69!/69!/0!*(0.99^69)*(0.01^0)
34.8% chance of obtaining one 1% item: 69!/69!/1!*(0.99^68)*(0.01^1)
12.0% chance of obtaining two 1% items: 69!/67!/2!*(0.99^67)*(0.01^2)
2.7% chance of obtaining three 1% items: 69!/66!/3!*(0.99^66)*(0.01^3)
80,000 Energy (why you'd exchange energy for crowns during a promotion, I have no idea, no clue why you're even implying you'd do that)
For the following prizes we'll just use the standard rolls * chances method because writing out all the chances here would be too cluttered:
Approximately 13 Crest/Costumes
Approximately 7 3UV tickets
Approximately 16 2UV tickets
Approximately 32 1UV tickets
And all the money that comes with selling those.
I've done the math and putting the rares at 2,000,000 crowns each and the Crest/Costumes at 200,000 crowns, the calculated value of the box is a little over 100,000 crowns.
@Hex
If you're right, I salute.
I'm probably going to get a box, just to test my luck. Other than that, I need the energy.
I bet no one would buy the boxes if you had to buy them directly. (Like, trade 1000 energy for a box or something.)
Or you could get the $20 thing, get two boxes, not get anything from either of them, then continue playing like a normal person with 7500 CE pocketed.