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Spoilers for the questions ahead.

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Interesting, but I can't believe 85% of people would get the first question wrong. That seriously blows my mind. That was the easiest question of the bunch. I found the center to take the most time to figure out, and the bottom is more opinion. Like the author says, it depends on if you can lose $100 or not.Never heard of it. I'm very typically out of the loop so it's no surprise. What is surprising is that 85% would still get it wrong since it's famous, as you say.

Another way of phrasing that first question to make it more clear how the bias can screw with you.

1. She's a feminist

2. She has had to deal with unwanted sexual advances at least one time in her life

3. She's a feminist who's had to deal with unwanted sexual advances at least one time in her life

Read those? What's your answer now?

The answer should still be the same. 3 is always going to be no more probable than 1 or 2. Two things at the same time is always less probable or exactly as probable as either of those things on their own.

I'm confused on what the bias is supposed to be. The information given points to her being a feminist "where she was an active volunteer in an advocacy group for women's health." I didn't see anything about her being a sanitation worker, so I ranked them 1, 2, 3. The reason is that, given the data, she's *most likely* to be a feminist and like you said, "x = true *and* y = true" will always be less probable than "x = true *or* y = true."

Supposedly the bias is that we give more credence to a more specific and credible-sounding description than something vague. In the book, Kahnemann asks three completely independent groups one of those three questions each and gets them to assign a likelihood. I think the '85% of people got this wrong' is an incorrect description of Kahnemann's experiment... it'd be more like "in 85% of cases, the probabilities assigned by independent groups gave an impossible result"

Interesting. I'm having trouble wrapping my head around the bias part. Are you saying that since the 3rd option is very specific, people are inclined to believe it? If so, I wonder why that is. Thanks for the information!

Not exactly because it is specific, but the specific details form a narrative that sounds plausible.

*The Black Swan* descibes Kahneman and Tversky asking forecasting professionals to give odds on the following two events:

a. A massive flood somewhere in America in which more than a thousand people die.

b. An earthquake in California, causing massive flooding, in which more than a thousand people die.

The first event was rated less likely than the second, even though that description includes the second scenario and more.

Another example:

*Joey seemed happily married. He killed his wife.*

This seems unlikely; it doesn't make sense.

*Joey seemed happily married. He killed his wife to get her inheritance.*

Now it seems more likely, even though we have reduced the possible scope of causes.

Thanks for the explanation! I think I was doing what cgod was doing on the 2nd question and looking at it too deeply, when it was a much simpler question than what I imagined.

I missed #2 - I said "U only"; didn't spend enough time thinking about it.

A rule that randomly assigns letters to cards would be as good an assumption as the odd even number assignment. There is no reason turning up two cards should be so illuminating. I'd guess there are many other rule assignments that could be giving in this 4 card system.

I don't really follow what you're saying here.

The article correctly points out that if you turn over the "9" card and it *does* have a vowel on the other side, then you've disproved the rule (which I should have realised).

[edit] To be clear - turning over "U" and "9" allow you to disprove the rule; turning over the other two cards tells you nothing about the rule at all.

You're right, but the question is asking about a specific rule and is asking you to assume that the four cards presented are an accurate representation of the rest of the deck. "If a vowel is printed on one side of the card, then an even number is printed on the other side."

Therefore, we can prove the rule by using two specific cards.

Flip over the "U" card because it directly matches the variables given. U = vowel. A vowel is therefore printed on the card, and thus the other side of the card must have an even number printed on it, else the rule is false. You can prove the rule absolutely false with this card because if the character on the other side of the card *isn't* an even number, the rule is false. You know this because this card has a vowel *without* an even number on the opposing side. However, if the character *is* an even number, you've only proven the rule true in this instance. There could still be another card that breaks the rule.

In order to unequivocally state that the rule is true, you have to then turn over a card with an odd number on it, because you've already proved that a vowel will be paired with an even number. Now you have to make sure that vowels will *not* be paired with an odd number as well. If they were, then the rule would be false. Therefore, you have to turn over a card with an odd number in order to determine what odd numbers will be paired with. Again, if it's a vowel, the rule is false and if it's a consonant, the rule is true.

The reason we wouldn't use the "J" card is because we don't care what consonants are paired with. The rule doesn't state a relationship between consonants and anything else, so we learn nothing by turning the card over.

The reason we chose "U" instead of "2" is because we need to be sure that an even number is printed on the other side of the "U" card. We don't necessarily care what even numbers are paired with as long as every vowel is paired with an even number.

- In order to unequivocally state that the rule is true, you have to then turn over a card with a number on it, because you've already proved that a vowel will be paired with an even number.

In general, this is not true. You must turn over the '9' to verify the rule *given these four cards*.
Turning over the "2" doesn't help, because it may have a vowel or a consonant opposite, and you learn nothing either way, since both are allowed by the rule.

[edit] In other words, you can *never* prove the rule, without examining all cards. The best you can hope for is to falsify the rule. Even if there were no odd numbers showing, you'd be in exactly the same position regarding the rule - it would be still be "true as far as I can tell".

I just forgot to put odd there is all. Should read "turn over a card with an odd number on it."

You can prove the rule, because as long as a vowel is always opposite an even number, you're good to go. The only other card you need is the 9, because it will tell you if the rule is true or false by whatever its opposite is. If it's a vowel, the rule is false. If it's not, the rule is true. You can then infer that the rest of the deck follows this rule.

Agreed; and now I see what you're saying.

I was misled because I was using the words "prove" and "verify" as different things, whereas you were using the word "prove" in the sense I was using the word "verify". Frickin' semantics!

I'm probably not using the words correctly in a mathematical sense. I was just trying to explain the problem the way I knew how. I'm taking Finite Math in the Fall, so I should probably brush up on the correct terminology. Oops!

Yea, that's what I'm saying. They didn't specify rules such that the information revealed by these four cards would allow a person to infer the rules for the whole deck but such rules could have been stated. It would be possible to disprove the rule but any number of convoluted rules could be applied to the deck that wouldn't hold for the proposed "rational" conclusion.

I am guessing the person who wrote the test left some conditions out.

The rule is given - you are not asked to infer any rule, only to confirm or deny the given rule.

You can confirm that the two cards you picked up conform to the rule but there are alternative systems of rules cause the two cards to be even and have vowels and not conform to the rules.

You are told to "determine whether the following rule holds for the deck" and that "these four cards represent the rest of the deck." I suppose that you are supposed to infer that if "these four cards represent the rest of the deck" and if vowels on the cards are even than "the rule holds true for the deck."

I just couldn't say that the rule holds true for the deck by turning up two cards, even if they conform to the rule, as long as there are other set of rules that could invalidate "all even numbers have vowel."

The author put "(assuming these four cards represent the rest of the deck):" in the problem. So you have to assume that the rules followed by these four cards do, in fact, represent the entire deck.

But there are different sets of rules that could govern the information that you glean from the four cards. The simplest rule could be that all the cards have been given random assignments. The four cards would still represent the rest of the deck. All kinds of rule sets could be represented in any four cards as long as you only get to see four cards.

We are have one given, "these four cards represent the rest of the deck" but at no time are we told that if "If a vowel is printed on one side of the card, then an even number is printed on the other side" than the deck must all be arranged such that ALL cards with a vowel must have an even number. There are other assignments that could hold true. I'm sure they meant it to mean that you can confirm that all cards with a vowel must have an even number which can be confirmed by turning up two cards but if you are putting up a logic puzzle than you should state you givens in a super hard way.

The second I see this kind of thing I get a the same frightened feeling that I got taking structured logic exams. Leave out a given that seems obvious and you fail the problem. If you turned up an even without a vowel you could toss this shit out but if you turned up even's with vowels you still can't confirm the rule.

I've read this question about 20 times now and I guess I'm just not reading it with good will. This is the kind of thing that teachers would hit us with in logic classes. If you take the premise that "these four cards represent the rest of the deck" means that numbers with vowels are even and you confirm it by turning up just two cards and there is no other assignment of vowels to numbers even or oddness that can exist such that this rule isn't true.

There are assignments of numbers and eveness that could hold for four cards such that cards with even numbers will have vowels but not hold true for the rest of the deck. It would in no way invalidate the premiss that "these four cards represent the rest of the deck" and still not make it true that all cards with even numbers have vowels on the other side.

I understand what you mean now. I think the test is a bit more basic than what you're looking for and sort of falls into its own little "box" of rules. It's more designed as a quick "gotcha" than a deep, winding path of logic, IMO.

She gives it to you to flip (which implies it is fair).

First one was easy. I got the middle one wrong but I only spent about five seconds thinking about it. The final question is broken to begin with. Heads side is technically heavier than tails (at least in America) I could not to take the bet because the realistic increased probability of tails is enough of a deterrent.

But you get to flip the coin, so you can put that to your own advantage. Besides, the bias in flipping coins is only around 1%, which is far less than the 2x difference between what you might win and what you might lose. (Incidentally, when flipping coins, the important factor is which way up the coin starts – weight is only more important when spinning the coin).

Interesting, I did not know that. Thanks for the info!

I got them all right; how did other people do?

There's no point flipping J over, because the rule doesn't say anything about cards which don't have a vowel on the letter side. There's no point flipping 2 over, because if it isn't a vowel then the rule doesn't specify anything, and if it is a vowel, then the number is even anyway, so the rule still holds. On the other hand, if 9 has a vowel on the opposite side, the rule is incorrect, so you have to flip it over to check. Likewise, if U has an odd number on the opposite side, the rule is also incorrect, so you have to flip it over too.