Why you should expect "bad" luck if you open <1,000 CL packs.
If you open some packs and ask "do I have great luck?" the answer is probably yes. If you open some packs and ask "do I have bad luck?" the answer is probably "no, you have average luck". But that's because the amazing packs are incredibly rare and stick in your mind, whereas bad packs are incredibly common.
I've also seen two posts in the past 24 hours by @seiimeii on hive and u/dt408 on reddit complaining about bad luck or losing value on packs. But again, the outcomes for these people are not bad luck, as I'll show, but normal luck that should have been expected and avoided.
So my main goal in this post is to compile some points I've made across various posts and help people think about the value (or lack thereof) of opening packs unless they open >1,000.
What is Expected Value and why should I care?
The expected value of something is what would the value of it be, on average, if we received it multiple times? For something like a fair coin, your expected value of 0.5 heads. If you flip a coin enough, you should get ~50% heads and 50% tails. Expected value gives us a good sense of what an average outcome is.
Importantly though, it says nothing about the chances of that average outcome. Back to the coin example, the chances of getting an average flip is actually 0%. You can't get 0.5 heads in a flip; you either get 1 heads or 1 tails. The expected value describes exactly 0 actual flips.
The other important part of this is it depends on a large number of trials. If I flip a coin twice, my chance of 2 heads or 2 tails is 50%. But if I flip 1000 coins, my chances of all heads or all tails is 0.5^(1000), which is vanishingly small. The expected value only described your actual expectation if you do a ton of trials.
Everything here applies to pack openings as much as coin flips. Splinterlands.com has a front page expected value of a pack. At the time of writing, a chaos legion pack costs $5.04 and has an expected value of $5.29. Someone not aware of what I've written about expected value above may think "oh cool, I can open a pack and sell it for more than I paid, ad infinitum, and then go and buy 10 packs worth a total of $10 (not each, total, which is not unlikely if you get 4 commons and a rare each time). So @seiimeii's strategy of buying and cracking packs to sell for a profit was doomed from the start.
Expected Value vs. Variance
So how do we know how what is a sufficient amount of packs? Obviously there is some personal preference here. I like to focus on the chase cards: what are the cards everyone wants? In splinterlands, the obvious answer is gold foil legendaries (GFL). Assuming you're using potions (you're using potions right?) your chance of a GFL in a pack is 1/1562 cards, or 1/312 packs. So if I open 312 packs, my expected value of GFLs is 1. But again, this is expected value and says nothing about your chance of actually getting one. The chance of getting 0 GFLs after opening 312 packs? 37%. So the reddit user complaining about no GFL in 340 packs? They had over a 1/3 chance of being that unlucky.
Below is a graph summarizing this. As you open packs, what is your chance of getting 0 GFLs? As you can see, until ~1000 packs, you have a decent chance of getting none. So anyone who complains about getting 0 GFLs after opening <1000 packs hasn't had really bad luck.
The value of some vs. many packs
Yes, GFLs are what everyone wants, but they're just one part of the picture. What about the actual value of packs? Below, I simulated opening 10, 100, 1000, and 10000 packs 100 times each and seeing what the average value of the resulting cards would be.
This box and whisker plot is pretty striking. If you open 10 packs, the variance is HUGE. Sure, you could get an average of >$20 per pack, but you could more easily get <$4. As you open more packs the variance goes down, to the point where opening 10,000 packs has very little variance.
What about FUN?
The counter to all of this is that cracking packs is fun. And that fun surely has some value. I've opened 2 packs for exactly this reason. If you value fun and enjoy seeing what's in a pack, don't let me or anyone else tell you otherwise. At the end of the day, this is a game and it should be fun. But if you get fun/enjoyment from earning value, balance that against the fun you get from cracking the pack.
Generally, don't open packs. Buy singles. You'll get the cards you want and save money
If you must open packs, open >1,000
If you're opening <1,000 packs, don't be surprised if you get 0 GFL
If you're opening <1,000 packs, don't be surprised if you lose money per pack on average.
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