The Maths Behind Loot Boxes - Are The PoE Carnival Boxes A Good Value Proposition?
70TLDW version - NO
Buying enough to get a full set will result in a lot more duplicates than most people intuitively expect. Not just a few more. A LOT MORE.
The mathematical methods here apply to any lootbox with publicly announced probabilities. If a lootbox in another game doesn't specify probabilities, you should assume they are hidden for a reason and the entire thing is a blatant scam.
On average, getting one of the complete sets (all Ringmaster rare pieces) will take 181 lootboxes and getting both (Ringmaster PLUS Harlequin) will take 227.
I've tried to avoid the ethical questions here to focus on the numbers and value proposition, although ethical questions about lootboxes are everywhere. However for full disclosure of any potential bias, I support regulations to ban lootboxes in games.
References and further reading for those with a maths background:
Wikipedia article on the Coupon Collector's Problem (recommended if you have a first year uni maths background):
https://en.wikipedia.org/wiki/Coupon_collector%27s_problem
Related mathematical journal article (highly technical, recommended only if you are active in the field or have more than a bachelor degree maths background):
https://arxiv.org/abs/1412.3626
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