# How reliable are intuitive answers?

Here are two examples I found in a book I’m currently reading.

1) Try to think of an intuitive answer without spending much time:

**A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?**

2) Again, try to think of an intuitive answer without spending much time. Does the conclusion follow the premises, yes or no?

**All roses are flowers.**

**Some flowers fade quickly.**

**Therefore some roses fade quickly.**

(Source: D. Kahneman)

Is this why Lehman Brothers went bankrupt? Were their people relying on intuitive answers?

What is your own personal experience with intuitive answers?

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## 75 Answers

Intuitive answer…

1 – The ball costs ten cent… *but only if purchased with the bat*.

2 – Yes, some roses fade quickly… *as has been my personal experience*.

*“What is your own personal experience with intuitive answers?”*

Generally, not very good. I’d prefer to study the situation before offering an answer. In doing so, considering the depth of any given situation, my intuitive answers often lack substance.

1. The ball costs five cents.

2. No, the conclusion does not follow from the premises.

Many intuitive answers *are* reliable, which is why we’ve evolved to have the intuitions we have. What has evolved, however, are heuristics: patterns of thought that are helpful in the most common situations, but that do not hold in all cases. They can be fooled, and they often are in exactly the cases where things start to get tricky. Still, that isn’t to say they aren’t reliable. They’re just not reliable enough to get us through situations for which they were not designed.

The ball costs 5 cents.

I don’t think that’s why Lehman went bankrupt, I think greed and cockiness were their problem.

@SavoirFaire Jinx. I think we both get a coke for our answer to the math problem.

@JLeslie – I also thought that greed was the main problem for Lehman, but I’m not so sure anymore. In addition to Kahneman’s book I’m also listening to Joseph Stiglitz’s “Freefall” audio book and he makes a point that most investment bankers actually believed that the creation of mortgage securitization instruments would be best for everyone. They really didn’t see the flaws in their thinking. Greed was part of the problem, but maybe not the main one. The way human brains work might be the real problem. Kahneman claims that many people are overconfident and prone to place too much faith in their intuitions.

He cites a study for the bat-and-ball problem: “Most people choose 10 cents. More than 50 percent of students at MIT, Harvard and Princeton gave this intuitive answer. At less selective universities, more than 80 percent get it wrong.” Link

The deception takes place in the ambiguity of the question.

The term *“the”* ball, misleads the answer to depicting *“the”* ball that came with *“the”* bat.

A more honest question would have stated *“a”* ball, as to distinguish it as a separate purchase from *“the”* ball that comes with the bat.

You can get any answer you want by asking the question in a particular way.

Don’t take me wrong… I understand that this question is designed to be misleading.

The logistics of math are only as reliable as the input. Even math can be twisted to mislead us into believing reality is twisted.

But in reality, we all know that retail stores offer discounts when items are purchased together. A single bat, and a single ball, would cost more than buying them together.

@RealEyesRealizeRealLies – Yes, misleading is the point here. Kahneman studies cognition. What I find amazing is that more than 50% of people get it wrong according to these studies.

Jellies seem to be much better than the average, though. So far that is.

If you haven’t read about him already I think you may like to read about Karl Popper and Critical Rationalism and The problem of induction

It appears to be 10cents when purchased with the bat to me Edit, no wait it cant be 10 cents can it as that makes the bat 90cents more expensive than the ball and also regarding the roses, from the information given you can not induce that any rose will necessarily fade, infact it is quite probable that roses do not fade quickly, also “quickly” is a vage period of time, to induce that some roses fade quickly it is necessary to clarify what is meant by quickly. Some flowers have been known to fade in minutes. Therefore the only true answer that can be given for that is via personal observation and falsification of the statement “All roses do not fade quickly whereby “quickly” is defined as x minutes”

edit: by the way thanks for this question, it has helped me greatly with revision for my upcoming exam in the philosophy of science and the problem of induction along with other topics…

I can’t accept there is a “wrong” in this question. I’d actually be more inclined to propose that the wrongness came from the way the question is presented, as to intentionally mislead.

@mattbrowne @RealEyesRealizeRealLies I think the reason people guess 10 cents is they don’t check their answers. Every time I do a subtraction problem, I add my answer to the smaller number to check myself. With math I always try to check myself by putting the answer back into the original problem if I am solving for an unknown. I guess in a way it is like saying I don’t rely on my intuition, but for me, with math, math is very different than other problems in life.

@mattbrowne I do truly believe those guys at Lehman’s believed the bullshit they thought and sold. I don’t know all the details of the Lehman’s situation, but generally I think it is amazing how people will ignore history. During this financial mess I am reminded of the US stock market crash of 1929, and all the times in history we have seen real estate tumble down. Just like never forget the holocaust, there should be a mantra for good things do end, there is no such thing as a sure thing, and if it seems to good to be true it probably is. Why very educated people who are old enough to know better believed the “good times” would roll on forever I will never understand.

@JLeslie The question didn’t ask how much *“a”* ball cost. It asked how much *“the”* ball cost. In this light, I believe ten cent is the correct answer.

Thanks for the reading tip, @bongo. I’ve read some work of Popper, but not the two books you mentioned.

@RealEyesRealizeRealLies if the ball is 10cents that makes the bat 90cents more expensive than the ball. Not a dollar. therefore the ball is not 10cents.

@RealEyesRealizeRealLies Right. The ball costs 5 cents. Do you agree *the* ball costs 5 cents now that you have seen the answer, or do you still question it?

Though it may seem that *“a”* ball by itself costs 5 cent, that math is based upon *“the”* ball that comes with *“the”* bat. We can be no more sure that *“a”* ball by itself costs 5 cent any more than we can be sure that roses fade quickly because they are flowers.

This question never mentions *“a”* ball by itself. It questions *“the”* ball that comes with a bat.

@RealEyesRealizeRealLies If the ball costs 10 cents and the bat a dollar more, then the bat costs $1.10 so that means the two together would equal $1.20.

*“If the ball costs 10 cents and the bat a dollar more”*

That’s not the question. So it cannot be extrapolated from *“the”* ball to *“a”* ball by itself.

@RealEyesRealizeRealLies If you are trying to say a ball by itself would cost a different amount than when purchased with the bat, then we would have no idea how much a ball by itself costs, because we would have no information for that from the question. It could cost 50 cents; why would 10 cents be correct? It implies to me you are trying to do the math, but not understanding how it all adds up. 10 cents was my gut reaction also, but then I checked the math and saw my error before answering. It’s easy to understand why most people answer 10 cents, it’s a brain teaser.

The question is leading to suggest there is one ball in the equation by stating *“the”* ball. But the answer requires two balls to be in the equation. A more proper question would state *How much does “a” ball cost.*

Sorry, I can’t see why this could be some kind of linguistic problem here. All bats cost $1.05 and all balls cost $0.05. Therefore a bat costs $1.05 and a ball costs $0.05. And the bat costs $1.05 and the ball costs $0.05 as well, when pointing at any in particular. Any given bat costs $1 more than any given ball. The sum of any given bat and any given ball is $1.10. Right?

@RealEyesRealizeRealLies I’m reposting the question here for simplicity:

**A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?**

How do you get two balls? A bat and a ball means one bat and one ball, do we agree? The two items equal $1.10, do we agree on that too?

To every thing there is a season. I am an intuitive person, but I also know when the situation requires detail then I either do the hard work or (If my intuition tells me it’s over my head) hire someone who is really good with the details. (My first instinct was that the ball was $.10, but my intuition told me that it was probably the wrong answer.)

I know the ball costs 5 cents because we had a question about it a while ago. :P

Intuition is like guessing, sometimes you’re right, sometimes you’re wrong. Not very reliable.

@Keep_on_running , I really think you’re confusing instinct with intuition. I believe that instinct is more like guessing, intuition is knowing. If you were wrong, it was instinct.

@Judi

They’re both in the realms of guessing. As with most decisions, you don’t know if it was the right one until after the event.

@Keep_on_running ; Most people with more analytical minds and less intuitive minds would agree with you. I think there is value in both.

My husband became a believer early in our marriage. There were a few instances where I couldn’t explain why, but I didn’t want to hire someone, or I just “knew” that we needed to make a different decision about something. I couldn’t explain why, so he went with his logic instead of my intuition. Every single time, my intuition proved right. Now we weigh both his logic and my intuition before making big decisions.

I think that makes us a good team. My intuition run amok could turn into a disaster, but him trusting my intuition, and me respecting his logic makes for a great combination.

@Judi

Sounds like you have a great relationship and your intuition has worked well for you. :) Our subconscious mind is probably more powerful than we think, I’m not quite sure I really believe in it yet though.

I think I am incapable of intuitive math. I actually think this is one of my strengths in math. I have long learned that what seems right on the outset is usually not right, and so I never make assumptions when I first read a problem.

Unfortunately I can’t properly answer your first question because I just saw that question recently and already know the answer.

My answer to your second question would be that it does not necessarily follow from your premises that some roses fade quickly. Again, I don’t think I really answered that intuitively. My brain just automatically did a sort of venn diagram and saw that roses and flowers that fade quickly don’t need to overlap.

My strengths lie in fields that so often defy intuition like math and physics, so it simply is not a mental habit of mine to make guesses.

@mariah, I don’t think people believe they are guessng when they say the ball is 10¢. They think that is the obvious answer. It is such simple numbers it is not something that has to be written down and worked out, it can be done in our heads. My first inclination was 10¢ also, but then, as @Judi pointed out, I intuitively figured there is some trick, and my math side is to always check my math, to actually fill in the answers and add it all up. I think more than guessing, it demonstrates how some people are willing to question their own thoughts. Scientists have a hypothesis and then set up an expirement to see if their idea is right. Most people just think they are right period.

They don’t think they’re guessing, but they are. They are choosing a number because it seems right without using any sort of mathematical process. I agree, “it demonstrates how some people are willing to question their own thoughts.” That’s a good way to put it.

I think of intuition as a muscle that you sort of train; and the more exposure/experience you have in a particular field, the more likely it is that you will be able to have a correct intuition about something. So in the case of us who paused to question the 10¢, that’s probably because we have had experience with these kind of trick questions before.

There are also sorts of fields you can be intuitive about: cooking, baseball, stock market…

What proof do you have that the ball was $.05 and the bat was $1.05 or that they even exist?

Next: In the absence of knowing the care given the roses vs any other flower that question can’t be answered, and I knew this quickly. Were the roses thrown on the ground, put in a vase of water or left on the bush? What was the care given the control flowers?

There is a reason I’m on your friend’s list, @mattbrowne!

Oh, I seem to have missed @ETpro’s question two months, ago.

I also think there’s a difference between instincts and intuition. The latter requires a significant amount of learning. One needs a concepts of numbers and how to add numbers.

@Dutchess_III – Why do we need a proof? The prices were the premise. Same for the flowers and roses. And the question wasn’t about what can happen to roses and care given to flowers. The question was whether the third statement was a logical consequence of the first two. Makes sense?

@Dutchess_III It’s a theoretical problem: we assume there are a ball and a bat that, together, cost $1.10. We are then told that the bat costs $1.00 more than the ball and asked to figure out what the ball costs. The only needed proof is as follows.

Let x = the cost of the bat

Let y = the cost of the ball

We know from the explanation of the problem that x + y = $1.10. We also know that x = y + $1.00. That means we can substitute as follows:

(y + $1.00) + y = $1.10

We can then drop the parenthesis because addition is associative. This means that:

y + $1.00 + y = $1.10

from which it follows that:

2y + $1.00 = $1.10

We then proceed to isolate the variable first by subtracting $1.00 from each side of the equation:

2y = $0.10

and then by dividing each side by the coefficient (that is, the multiplicative factor that is being applied to the variable):

y = $0.05

Since y is the cost of the ball, we now know that the cost of the ball is $0.05. And since the cost of the bat is x, which is equal to y + $1.00, we also know that the cost of the bat is $1.05.

I thought @Dutchess_III was joking anout the proof. Just making fun?

It is also interesting (and informative) to note how far people will go to defend their first impressions. Whether on purpose or not, @RealEyesRealizeRealLies’ answers are quite instructive here. For it obviously makes no difference if the question is worded as follows:

*A bat and ball cost $1.10. A bat costs one dollar more than a ball. How much does a ball cost?*

People will still give the same incorrect answer much of the time and then try to justify it later. This sort of *post hoc* argument is something that psychologists have observed many times. The original wording, after all, is not really ambiguous at all. It is standard English to talk about “a bat and a ball” and then to subsequently refer to those objects using “the” to signal that we have *not*, in fact, moved on to other objects.

And despite protests above, “a” actually *does* follow from “the.” If there is a unique object to which “the” refers, there must also be *an* object. There can be no unique object if there is no object at all.

@JLeslie Just for fun. Might as well have the proof somewhere on this thread anyway.

Thank you @JLeslie!

Oh…wait. I see it now. I see why the bat would be $1.05….how interesting!

X=.10, Y=X+1.00.

X+Y=1.20 because

.10+1.10=1.20.

:) Can’t wait to throw that down in the class room!

@SavoirFaire I’m being argumentative to illustrate the linguistic paradox of following this question through to its ultimate confrontation.

Yes, when put into equation form, the bat costs $1.05 and the ball costs $.05. I have no problem with that.

Let’s go with that, having established the bat as a $1.05 item. That will defeat the original premise of the word problem in every day retail transactions. For example:

If the ball has been established at .05, and I place it on the register for the clerk to ring up, and I see a bat available at the point of purchase, I’ll ask the clerk *“How much for the bat”?* If the clerk says *“one dollar more”*… How much will you give him?

That’s all I’m saying… is that in word form, the question is (or at least can be) misleading, as it is designed to be in this situation. But as a pure equation, the answer is obvious.

And yes, if I am to refine my skills at selling ice to eskimos, then I’ll attempt to spin anything I can to my advantage… *which isn’t much different than the original intentions of this Q*… to spin it with the hope of achieving a particular result.

@RealEyesRealizeRealLies I see your point, but I can’t see that happening in a retail situation, plus the original question says the two equal $1.10. In your scenerio the bat and ball would cost $1.05 for both, so your original answer still doesn’t make sense, even with your explanation immediately above.

In a retail situation, the equation would already be completed with price tags on the objects. This question assumes the items are priced as a set from the very beginning.

@RealEyesRealizeRealLies There is no linguistic paradox. It’s just a simple mathematical problem that most people get wrong the first time they look at it. Your example with the clerk makes no sense and is just more *post hoc* justification: he would either say “one dollar more than the ball” (which is the same as how the original question is worded) or ”$1.05.”

But the question isn’t about retail situations. It’s about mathematics and the critical thinking skills that underlie mathematical reasoning. What grounds do you have to insist that the question was *designed* to be misleading other than the fact that you got it wrong? For as far as I can tell, the question is neither misleading nor attempting to be misleading. It’s just got an element that’s easy to miss.

Here’s another one for you:

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take

100 machines to make 100 widgets?

As to the second problem, its linguistic form is also misleading. It is close enough to the proper form to cause one to believe it is a proper truth statement, when in fact it is not.

The proper form is based upon:

All men are mortal.

Socrates is a man.

Therefor, Socrates is mortal.

All roses are flowers. (should read – Some flowers fade quickly)

Some flowers fade quickly. (should read – This flower is a rose.)

Therefore some roses fade quickly. (should read – This rose may fade quickly)

If it takes 5 machines 5 minutes to make 5 widgets, how long would it take

100 machines to make 100 widgets?

Owning a print shop I understand that it still only takes five minutes. I’ve got a dozen printers. My intuitive answer is influenced by experience. Which is what I was going to bring up earlier, introducing an element of chaos into the equation. I decided to let it go.

@RealEyesRealizeRealLies The second form isn’t misleading, it’s invalid. That’s the point: the question tests whether or not you are capable of distinguishing a valid argument from an invalid argument. If you’re going to complain about it being misleading, you might as well abandon all responsibility for anything. It may be wrong for politicians to lie to us, but it is also prudent for us to know how to figure out when they’re doing it.

As for the widgets question, you are correct. I don’t get your point about chaos, though. What claim do you think this affects? No one denies that experience influences intuition, after all, or that different people will have different intuitions. That’s one reason it is good to know how to follow the proper steps rather than only having intuitions. Remember: @mattbrowne‘s original question was “how reliable are intuitive answers?”

@SavoirFaire *“That’s the point: the question tests whether or not you are capable of distinguishing a valid argument from an invalid argument.”*

Completely agree.

@SavoirFaire *“If you’re going to complain about it being misleading, you might as well abandon all responsibility for anything.”*

Completely agree. However… complaining about it is taking responsibility, not abandoning it.

@SavoirFaire *“I don’t get your point about chaos, though. What claim do you think this affects? No one denies that experience influences intuition…”*

In this case of the widget, my experience is chaos, of which the questioner had no input for determining the likelihood of my intuitive response, *which is the real purpose of the question*. They don’t care what the answer is. They care why people give the answers that they do. I have no doubt there is enough data on intuitive answers for this question to calculate statistic on how this question is answered intuitively. Yet the deeper question of why the question is answered statistically remains unattainable unless an element of chaos (experience) is accounted for.

@RealEyesRealizeRealLies Complaining isn’t taking responsibility if you try to blame the question for your mistake after the fact (as with the ball and the bat).

As for the bit about chaos, I agree that people care *why* people give the answers they do. I disagree that the question was designed for any specific purpose, however. My understanding is that the question existed, someone noticed that most people get it wrong, and *then* people started looking into why. So you seem to have the teleology backwards. Experience is already understood in the literature as part of the reason. Again, then, I don’t see what claim you think the point about chaos affects (unless it is just redundant because you had a mistaken assumption about the research).

I took @JLeslie‘s comment *“it’s a brain teaser”* as meaning these questions are particularly designed to initiate discussion about intuition, and not because they necessarily represented any previously known phenomenon of consistently wrong answers.

At least that’s what my intuition told me.

As @Judi suggests *“My first instinct was that the ball was $.10, but my intuition told me that it was probably the wrong answer.”* Could it be that she came to this conclusion because she thought the question was designed to be deceptive in the first place? Doesn’t mean that it was actually designed that way, but I propose these problems are now propagated on the web to get certain answers which prove the point they were posted for.

@RealEyesRealizeRealLies I’m not sure that @mattbrowne was trying to make a point as much as ask a question and get us to think. I don’t really think that the questions are misleading, though. They are quite straightforward. It’s only when we try to take shortcuts—to solve them through intuition rather than by figuring them out step-by-step—that we make mistakes. We mislead ourselves; the questions do not mislead us.

I understand and may actually be swayed to agree with you in this case. But then, is there any such thing as a *“loaded question”?*

@RealEyesRealizeRealLies Yes, there are loaded questions. The classic example, of course, is “are you still beating your wife?” It is a loaded question because it contains a presupposition designed to cast even the correct answer in a negative light. One can only answer loaded questions by reframing them—e.g., by saying “I have *never* beaten my wife.” The questions in the OP are not loaded. We are simply prone to tripping over them.

@RealEyesRealizeRealLies I just want to see your math, you never showed us math as to why your 10¢ answer is correct. You just keep going around about the wording. Show us how your math fits the equation and is a valid answer.

We have provided our math, it is a mathematical question. Here it is again:

**A bat and ball cost $1.10. The bat costs one dollar more than the ball. How much does the ball cost?**

Put in your 10¢ and make it work? It doesn’t. It is a very black and white question that has an answer. If you want to argue that linguistically it is not clear then that is a communication problem, but I have to say I think you would be in the minority of not understanding. Getting the question wrong at first is not the same as not understanding, it is again a brain teaser meant to trick people who answer quickly. And, I realize you do understand our answer, i just use understand as a word to describe a breakdown in communication which can be the person writing the question or the reader. The intention of the question is to consider one bat and one ball purchased together equals $1.10 and the bat costs $1.00 more than the ball. We know it is the intention of the question, because the person who made it up is looking for the answer the ball costs 5¢.

Can someone also mathematically work out ”*If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?*”

I’m not entirely sure if my working out is correct.

@Keep_on_running Seems like it would be 5 minutes. Simplified; one machine takes 5 minutes to make 1 widget. So, It does not matter how many machines, they each make 1 widget in 5 minutes. Does that make sense?

Yeah it does, but how would you put that into an algebraic equation? I keep getting machine = 1, widget = 1, time = 1, whereas time should be 5.

widgets completed = machines * widgets completed per machine per minute * minutes

You know this formula is right because the units cancel down to widgets completed = widgets completed.

5 = 5 * x * 5

1/5 = x

widgets completed = machines * widgets completed per machine per minute * minutes

100 = 100 * 1/5 * y

5 = y

I thought the ball would cost ten cents so I must be ignorant.

Sorry @JLeslie, there is no magic math I have to allow for the ball to be 10¢. My answer is based purely upon arguing with a sales clerk who tells me the bat is $1 more. If that’s what he says, then that is what I’ll expect to give him. I’m wrong on all accounts, I know.

@JLeslie

*“it is again a brain teaser meant to trick people”*

*“the person who made it up is looking for the answer the ball costs 5¢.”*

So which is it? If everyone answered correctly, then these *“teasers”* would not get the publicity they do. I can accept they may not have been designed originally to *“trick”*, but they are indeed now published in order to initiate discussion.

As a purely mathematical equation, there is no confusion. That can easily be accomplished at the store with price tags. None of this would ever be put into word form unless the items were not tagged with prices in the store. The problem depends upon a conversation with a sales clerk. And if the clerk uses the exact same language as the question does, then how much will you expect to give him?

@Mariah Ah that makes sense, thank you.

@Aster You’re not ignorant! On the face of it that is the logical answer. However, if you put it into a math problem it puts it into a completely different light.

X=bat Y=ball

X=(Y+ 1.00)

.....^^^^^^^ that right there is the secret.

If you just plug the numbers in for X and Y and do the equation you’ll see it.

@RealEyesRealizeRealLies I guess since you do not supply the math, there is no way mathematically to make your answer work. Your 10¢ simply cannot equal the $1.10 and so your answer is wrong. Even if part of the question can be interpreted two ways, as in your example “the bat is a dollar more,” that way of interpreting the question cannot give you a $1.10 so then it is obvious the other interpretation that is the correct one. You can argue a communications problem, but it does not mean the question is worded incorrectly, only that it might seem ambiguous to some people in how they read and comprehend the question.

When I was in MX my MIL called for a taxi for my husband and I to go downown, the man said it is $25 one way, and $5 more for the return. When we got back she thought her bill would be $30, but it was $55. This is similar to your example. If the gentleman on the phone had say, so the total will be $55 round trip, she would have instantly realized the miscommunication and incorrect assumption she made from how he worded it. We have the total given to us in the bat and ball problem, so there is only one correct answer.

The brainteaser is not trying to be evil, just a trick that makes people think twice. Like what weighs more a pound of feathers or a pound of bricks.

You are fighting here for something I do not understand. Why is it so important for you to be right? That is how it is beginning to feel to me is you just have to be right.

I don’t know how evil came into the picture, but it’s an interesting way to phrase a defense. And I’ve already admitted to being wrong *“on all accounts”*, so the importance of being *“right”* is not in this equation… Unless of course, how you *“feel”* is an intuitive answer which prevents the math from being realized.

@RealEyesRealizeRealLies I just use evil as meaning purposely trying to trick people into something bad for them. I meant it is not such a serious matter, the brainteaser is a game, a riddle.

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