From the SWU Harvest Team – One of our goals here at SWU Harvest is to provide information about a pastime we love to others who also love it. Sometimes, we may not have the arithmetical acumen to properly picture the prime properties of percentages. In cases like this, we ask another member of the community to step in and help. That’s right: we classed up the place with GUEST AUTHOR PHILLIP RUDY!
Image credit: 20th Century Fox / Lucasfilm
Hi everyone! I’m Phillip Rudy, and I’m thrilled to be a guest here on SWU Harvest to talk about everyone’s favorite subject: math.
Once upon a time in a galaxy far, far away I was a nuclear-trained officer in the U.S. Navy. The only thing I love more than Star Wars cards is mathematical formulas about Star Wars cards. I was an avid player of the Star Wars Living Card Game back in the day but have largely stayed away from TCGs, though I admit to spending much of my [parents’] money on the old Decipher game. I’ll never forget how good it felt when I opened a pack of cards that my parents put in my stocking on Christmas to pull a Death Star. So, inspired by that rush at pulling the thing you want the most from a pack, I’ve decided to ruin everything by boiling it down to cold, hard numbers (jokes, jokes! I was told to keep this PG-13 though, so I’ll try to avoid any R-rated math jokes 😁). I’m so excited for this game. I just think this is an excellent opportunity to share a little bit about how powerful it can be when you understand how to reason quantitatively, and I hope to inspire you to try to add a little more math to your day.
Image Credit: Walt Disney Company / Workman Publishing Group
Math Complements Star Wars TCG Nicely
It is tempting to think of probability in terms of “How likely am I to get what I want each time I try?” In fact, what you should consider is the opposite of that, called the Complement, which is the probability you WON’T achieve the desired result in any given attempt. So rather than say the probability (let’s call that p) of me pulling a legendary card is 1:8, the complement is to say the probability, let’s call that q, is actually 1-p or
q=1-p
q=1-( 18)
q= 78
Why is the complement valuable? I’m glad you asked. Using q, we can write a function that will tell us what the probability of a desired result is after a given number of attempts by raising p to an exponent equal to the given number of packs opened, let’s call that N. Continuing with the above example using the p-value of 1:8 for legendary cards:
qafter_n_packs= qn=78n
So, applying this: if you open six packs for a sealed tournament (say at the pre-release!), about half the players will pull a legendary.1
qafter_6_packs=786
ppre-release_legandary=1-qafter6packs
ppre-release_legandary=1-0.45=.55 x 100=
55%!!!
Pop quiz: What’s the complement of that?
Jump to Hyperspace (Palpatine)
If, much like Chewbacca, you find yourself wanting to hit something with a spanner when you try to figure out just how many packs you’d need to open to get that sweet, sweet Showcase Palpatine: fear not! We can use what we just learned above to figure it out!
Let’s start with some baseline assumptions:
- The stated probability of getting a showcase card is 1:12 boxes.
- There are 24 booster packs per box.
- Therefore the odds of a showcase card are approximately 1:288.
Can you guess the steps if we use what we learned above? I’ll have to… compliment… you if you said solve for q.
p=1288
q=1- 1288
q=0.9965
NB: I am using four significant digits here because of the very, very small percentage.
We’re going to use the same idea as the next step above but with a slight twist that we’re not going to solve for a discrete number. I’ll explain below but first I’ll set up the equation like we did before. I’m going to call the chance you won’t get a showcase card after opening N number of packs q(n):
q= 1288
qn= 1288n
With me so far? You’re doing great, kid, but don’t get cocky.
We’re going to take what we just did and instead ask ourselves this: how many packs do I need to buy before I get a showcase Palpatine leader card? To answer this question, we are going to graph the function we just solved for and find a value of N, the number of packs we must open, that outputs a given value of p that we accept (e.g. I want a 50% chance that I will get a showcase Palpatine in the batch of packs that I buy). You can see from this graph that the probability asymptotically approaches 1 (100% chance) as you buy about 850 to 900 packs.
(NB: the graph below is in a logarithmic scale because of the very large X value.)
That’s 38 boxes (assuming we want about 900 packs), or at $4.99 per pack, that’s about $4,500 to guarantee you get that Showcase Palpatine.
The good news is, you might get lucky enough to pull the card you want after ten packs ($49.99) or maybe even a hundred ($499.99), so this is just a thought experiment. In no way should you use these back of the napkin calculations to do anything in the real world with real money.
I think that’s probably enough math for now, but if you’d like me to crunch some more numbers for you between now and March 1st, let me know in the comments, and perhaps I’ll be able to put out a companion piece to this one. Until then, may the odds be ever in your—no, wait, that’s not right—never tell me the odds!
- This is a simplified probability using two significant digits, which does not account for the 1/50 chance that a pack will contain a foil legendary, which is independent of the baseline legendary probability but if you’d like you would simply follow the same steps above replacing the 1:8 probability with 1:50 and then add to the 55% probability I calculated above. ↩︎
TCG booster packs, as with loot boxes, are like gambling and can become an addiction. Addiction is a sickness and can be treated. If you or someone you know is struggling with a gambling addiction, help is available. The National Council on Problem Gambling provides a range of resources, including answers to commonly asked questions, a gambling behavior self-assessment, information about treatment and the National Problem Gambling Helpline (1-800-GAMBLER) to help connect you with local resources or visit https://www.ncpgambling.org/help-treatment/. There is no shame in asking for help. Please take care.
About the Author: Phillip is an avid lover of card games, Star Wars, and Star Wars card games. He formerly served in the US Navy as a submarine officer, and in his spare time, he does law stuff to support his board game purchases. Phillip lives in New York City with his son, a pit bull, and two very judgemental cats.
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