What's The Deal With The Wheel of Fortune in Balatro?
Recently I, and many other people, have gotten absolutely infatuated with Balatro. The only way I've been able to describe it to my family is that it's a poker, roguelite, deckbuilder. You start a run with a standard 52-card deck, and as you make your way through your unique run, you get to customize your deck using various different types of items.
This isn't a review of the game-- that's something I'll probably get into in another post. This is a post talking about the Wheel of Fortune in Balatro. The Wheel of Fortune is an item that you can obtain and use, and it's part of the category of Tarot Cards in the game. On the Fandom wiki for the game, it states that the Wheel of Fortune, when used, has a '1 in 4 chance to add Foil, Holographic, or Polychrome edition to a random Joker.'
I assume since you're reading this, you know what that means. If you don't, Jokers are cards in the game that do a variety of different effects. Usually, I grab the ones for adding to my multiplier. If your Joker card(s) have the Foil, Holographic, or Polychrome effects, this adds on another benefit to what they already give you. Bottom line: they're good, and can add that oomph that you need to your run.
I've seen a fair amount of people complaining that the odds for the Wheel of Fortune to give you a buff seem way smaller than a 1/4 chance. For me, it's seemed to mostly line up with 1/4, so I wanted to put my statistics knowledge to use and write up a post debating it.
Statistics
A 1/4 chance is small. It isn't impossibly or incredibly small, but it is small. If a 2/4 chance means it happens .5 of the time (50%), a 1/4 chance means it happens .25 of the time (25%). A way we can think of this is by rolling a fair 4-sided die and hoping for a particular number. While our odds of getting, say, a 3 will be better than they'd be on a 6-sided or 20-sided die, it's still a 1/4 chance you land a 3 rather than 1, 2, or 4.
In addition, as far as I know, these are all independent events. If you roll a dice, then roll another one, what you got the first time doesn't affect what you'll get the second time. This is different from, say, pulling a card and not replacing it. If you pull a card and you don't replace it, your sample space gets smaller, and odds change. This isn't the case with dice.
Getting that 1/4 doesn't mean that you'll get one hit if you try 4 times, it means that every time you try, you have a 1/4 chance of making it. A lot of the time, we may just keep not getting that 1/4. We also probably aren't doing this hundreds or millions of times while logging it all, so the Law of Large Numbers doesn't help much in our belief of the odds. We can take comfort in knowing that the more trials we do, the amount of times that we did hit will get closer to 1/4. This may take a while, though.
Confirmation Bias' Role
So, what happens if we do the Wheel of Fortune and keep getting nopes? Confirmation Bias is a term in psychology, where someone will interpret or seek out information that agrees with their beliefs or worldview. If you keep getting nopes, you'll believe that it has to be broken, and that there's no other possibility. If you keep getting hits, you'll believe that everyone is overreacting and the odds have to be higher than 1/4. When you hit, but believe it's broken, you'll be more likely to blame it on stupidly good luck. This works in vice-versa, too. If you keep getting hits, when you get that nope you'll be more likely to brush it off as bad luck.
The most likely scenario is that we get more nopes than we do hits. The fraction 1/4, or the percentage 25% seems like way better odds than rolling a 3 on a 4-sided die or a 25 or lower on a 100-sided die. If you do it, and believe you have better odds than you actually do, when you fail you'll be inclined to believe that it has to be bugged and there can't be any other explanation. While it's true that the game might be busted, what I've witnessed correlates with how I understand it from a statistics point of view. Another way to understand it is getting a hit is the same odds of flipping a fair coin 4 times, and getting only 1 heads.
Conclusion
Probability has funny ways of tricking us. There's a reason why we put things into graphs: it makes data easier to read. Raw data looks confusing, and probability itself is at least a little confusing to most people. 1/4, .25, 25%: these are all easier ways to read the chance of something happening. They also look a lot better, or more likely, than actually writing it out. Odds of getting 1 head in 4 fair coin flips? 1/4. Odds of rolling a 25 or less on a 100-sided die? 1/4. It's important to properly process the odds that we're faced with, lest we come out proclaiming that it's broken. While it may be broken, I'm not inclined to believe this compared to just relying on what I know due to statistics.
I'm also going to write a game review for Balatro as a whole, so I'll link that [here] when it's done.
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