Daniel Kahneman – A Bat and a Ball Cost $1.10

Daniel KahnemanRIP Danny. Thank you for your insights and stories. I’m glad of the opportunity to know your work. You made a difference.

[Update 9/8/2010 A paper Danny co-authored, “Income’s Influence on Happiness” has just been released.]

… the bat costs $1 more than the ball. How much does the ball cost? Daniel Kahneman KNOWS that the first thought that entered your head was $.10–even if you’re a Computer Science major at MIT. But that’s the wrong answer.

Daniel Gilbert’s “Stumbling On Happiness” led me to Nicholas Taleb’s “Fooled By Randomness“. Both books cite the work of Danny Kahneman. I blogged a bit about him here. I have been rummaging around the internet looking for whatever I can find on Danny and his work and have come up with some excellent content. But let me give you a taste of the sort of fascinating facts you’ll hear in Danny’s lectures first.

In a study Danny (I don’t know him personally but after listening to all these lectures, I feel as though I do. He could no doubt name the cognitive bias this suggests) mentions in one of his talks, people are asked how much pleasure they derive from their car. They are then asked enough questions about the car to determine its blue book (resale) value. It turns out that there IS a correlation between the amount of pleasure the subject reported and the dollar value of the car. i.e. Yes, that late model BMW in the garage DOES give you more pleasure than my 20 year old Honda would. BUT! They then go on to ask the subject if they find their commute to work pleasurable, and guess what?– nobody does!. It turns out that the ONLY time people derive pleasure from their car is when they are THINKING about it.

From Wikipedia:
With Amos Tversky (Kahneman’s longtime research partner, with whom he would have shared the Nobel prize had Tversky not died in 1996) and others, Kahneman established a cognitive basis for common human errors using heuristics and biases (Kahneman & Tversky, 1973, Kahneman, Slovic & Tversky, 1982), and developed Prospect theory (Kahneman & Tversky, 1979). He was awarded the 2002 the Nobel Prize in Economics for his work in Prospect theory.

Major Contributions:

  • anchoring and adjustment -describes the common human tendency to rely too heavily, or “anchor,” on one trait or piece of information when making decisions.
  • availability heuristic -where people base their prediction of the frequency of an event or the proportion within a population based on how easily an example can be brought to mind.
  • conjunction fallacy -when it is assumed that specific conditions are more probable than a general condition that contains the specific condition. (i.e. You think you’re MORE likely to die in an air disaster brought on by a terrorist event, than you are to die in ANY kind of air disaster).
  • framing (economics) -reversals of preference when the same problem is presented in different ways. (10% fat vs. 90% fat-free!)
  • loss aversion -the tendency for people strongly to prefer avoiding losses than acquiring gains. (Why New Yorkers stay in New York for the culture, and Angelenos stay in LA for the weather!!).
  • peak-end rule – we judge our past experiences almost entirely on how they were at their peak (pleasant or unpleasant) and how they ended.
  • prospect theory -how people make choices in situations where they have to decide between alternatives that involve risk.
  • reference class forecasting -predicts the outcome of a planned action based on actual outcomes in a reference class of similar actions.
  • simulation heuristic – people determine the likelihood of an event based on how easy it is to picture mentally. (Why we buy lottery tickets.)
  • status quo bias -in other words, people like things to stay relatively the same.

Media – Most of these lectures have a fairly long-winded intro. Skip ahead if you don’t need the background info.

Explorations of the Mind – Well-Being: Living and Thinking About It. (YouTube)

Conversation With History – Intuition and Rationality. (YouTube)

Conversation With History – Intuition and Rationality. (Audio)

Explorations of the Mind – Intuition: The Marvels and the Flaws. (YouTube)

Nobel Prize Lecture. (YouTube)

Update March 2009- Kahneman and Taleb on the same stage discus the crash. (YouTube)

Update March 2010. From the February 2010, Ted Talk Daniel Kahneman: The riddle of experience vs. memory. (YouTube)

Found a few more DK links.
A Perspective on Judgment and Choice 24pg. PDF doc. on the subject of his Nobel Prize.
The Allais Paradox Wired magazine 10/10 (Archive)


12 thoughts on “Daniel Kahneman – A Bat and a Ball Cost $1.10

  1. Ronad

    I have been trying for a simple explanation of the “bat and ball” and I just don’t get it. The sound you hear is a gun going off!

  2. JohnH Post author

    The bat costs a dollar more than the ball. Working backwards from the $1.10. Take the dollar away you have ten cents. But if the ball costs ten cents then there’s nothing left over for the bat! The bat would cost nothing. Aha! So if we split the ten cents in half, 5 cents each, then when we add the dollar back we’re up $1.10.
    Hope that helps!

  3. Christine Draper

    See, if the ball costs ten cents (0.10), and the bat costs a dollar more (1.10, then a bat and a ball cost $1.20.

  4. Kevin

    I understand the intuitive answer of 10 cents is incorrect because then the bat would cost nothing. However I don’t understand why the answer is automatically 5 cents. Are we restricted to nickels? Why can’t the ball be 9 cents or 1 cent or anything in-between?

  5. John Post author

    It’s because the initial condition is that the bat costs a dollar more than the ball! If the ball cost 1 cent, then the bat would cost $1.09 which is $1.08 more than the ball.

  6. Wally Hammer

    The opening statement that the bat and ball cost$1.10 sets up the understanding that the bat and ball were purchased together and cost $1.10. So, if that is case as it was stated to be then the answer is .10 cents. What other scenario could be set up to create the 5 cent answer?

  7. Murray

    This is just to help convince some of those who seemed to think that there was something wrong with the question or answer. (Especially Wally Hammer)

    I’m going to start from the other direction to previous explanations, and show through working through the possibilities that the answer is that the ball can only cost 5c.

    We are told that the bat costs $1.00 more than that ball. I’m going to start purely with that stipulation and see what the consequences are. So let’s consider potential prices of the ball ($ball) and what that would mean for the price of the bat ($bat) and the total cost ($Total).

    $Ball + $Bat($1more) = $Total
    0c $1.00 = $1.00
    1c $1.01 = $1.02
    2c $1.02 = $1.04
    3c $1.03 = $1.06
    4c $1.04 = $1.08
    5c $1.05 = $1.10
    6c $1.06 = $1.12
    7c $1.07 = $1.14
    8c $1.08 = $1.16
    9c $1.09 = $1.18
    10c $1.10 = $1.20

    From the above, “10c” is clearly the wrong answer. It makes the total cost $1.20, not $1.10. Indeed, any cost higher than 5c for the ball must be wrong, as the total cost will be more than $1.10. Ball prices less than 5c mean that the total is less than $1.10.

    A price of 5c for the ball and $1.05 is the only scenario in which the bat costs exactly $1 more than the ball, and the total is $1.10.

  8. Adam C. Scott

    Or we just go back to using algebra.


    We have Equation 1:
    “A bat and a ball cost $1.10”

    b+a=1.10 (1)

    We have the 2nd Equation:
    “the bat costs $1 more than the ball”

    b+1=a (2)

    From Equation 1, we solve for b:

    b=1.10-a (1.1)

    Substitute (1.1) in to (2)
    (1.10-a)+1=a (2.1)
    Solve for a:
    0.10=a+a (2.2)
    0.10=2a (2.3)
    0.10/2=a (2.4)
    a=0.05 (2.5)

    From 1 we can substitute a back in:
    b+0.05=1.10 (3)

    And solve for b:
    b=1.10 – 0.05 (3.1)
    b=1.05 (3.2)

    So the bat is $1.05 and the ball is $.05.

    But the guy with street smarts may say hey not so fast. Why in the world would someone sell a bat and a ball for only $1.05!? There’s something fishy here.

  9. Jane

    Murray, THANK YOU so much for your explanation! Yours is the one that finally made it click for me to be able to understand this very simple yet very complicated problem. I was racking my brain non stop all day trying to figure out HOW this equation could possibly make sense. Now I can rest easy haha

  10. Mel

    Adam C Scott you are the only one who managed to help me understand it, purely because you wrote that the bat costs 1.05!

  11. Bill

    Kahneman uses this example to talk about our propensity to use what he calls System One. System One is the automatic part of our decision making and is effortless. It is the part of our brain that served up $.10 as the possible answer. Almost everyone has $.10 pop into their brains. Comment #4 above is a great example of our propensity to stick to System One rather that use the more effortful System Two where we have to think things through thoroughly. The key point is that we all make errors when we rely on System One in circumstances when System Two should be used, and we all do it.

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