Here’s a candidate for a question to illustrate a couple of related biases:
Given the following two dice roll records:
1: HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
2: THTTHTHHTHTTHHHTTHTHTTHHTHHTTTH
Which of the following is true:
A) 1 is more probable than 2.
B) 2 is more probable than 1.
C) Both are equally probable.
Now, I predict that there will be at least 1 “normal” person who answers C.
“Unbelievable,” you say?
Stay tuned!
I will make a stronger prediction: If this question were posed to 1000 randomly selected, well-dressed, Nordic-looking people found purposely walking the downtown sidewalks during daytime in a large American city (with luck, eliminating the possibility that I cheat by selecting 1000 people from insane asylums or from people who know no English), I predict that there will be at least 1 person who answers C.
Why? Because it is a well known fact that there exist, in much larger numbers than 1 in 1000, people capable, willing, and even eager to use the “toilet paper tube fallacy”. Any of such people combined with any of those who are susceptible to the “literalist fallacy” will answer C.
Let me make a stronger prediction. Even given a 4th choice, so slyly left out:
D) Beats me.
I predict that, still, at least one person will select C.
Now, list the following in order of probability:
a) That one person is a moron.
b) That one person is a computer programmer.
c) That one person is a card shark.
d) That one person believed that choice B was to be taken literally. That is, that B really (really!) means that the very first coin flip came out tails—NOT HEADS! - tails, the second heads, the third tails, and so on.
e) That one person ignored as much context around the dice roll question as he could. That is, that person pretended he was similar to a computer in seeing the world through what amounts to a toilet paper tube. Just the facts, Ma’am.
f) That one person is a card shark and a computer programmer.
g) b and c
h) d and e and f
i) All of the above.
“h”, anyone? :)
But, a thought on this question: How to avoid the conjunction fallacy?
Perhaps a better way to do so than keying on the word “and”, (which, as we all know, means “OR”, but not “OR and not AND”) is to key on the word “probability”. That is, when you see that word (or sense its meaning) as a goal, hand the question to the modern equivalent of a four-function calculator and let it grind out the numbers. To do so otherwise would be like multiplying 10821 by 11409 in your head, wouldn’t it?
d) That one person believed that choice B was to be taken literally. That is, that B really (really!) means that the very first coin flip came out tails—NOT HEADS! - tails, the second heads, the third tails, and so on.
I’m sorry—I suppose I’m probably missing something, but I can’t think of any other possible way to interpret this question. I agree that it is far more probable to see a sequence equally containing both heads and tails than one containing only heads, but it seems like you are asking for the relative probabilities of two highly specific sequences of the same length. Could someone please explain?
A. There is significantly greater than a 1 in 2^31 chance that the coin is significantly biased towards heads. This sequence overwhelms almost all priors of fairness, and thus we can conclude that the coin is almost certainly biased towards heads.
He’s rolling a die. As such, both “possibilities” are overwhelmingly improbable, as I have never seen a die labeled with heads and tails, and I spend a lot of time around dice.
Tabletop RPGs often use the term “roll M N-sided dice”, or “MdN” for short, to mean, “generate M high-quality random numbers between 1 and N”. The dice themselves are merely an implementation detail; they could be physical dice, or some random-number generator built into a collaborative RPG software program, etc. It’s common to refer to coins as “d2″s, because that’s the function that they serve.
Another interesting die roll that comes up quite often is “Md3”; the 3-sided die is usually implemented by taking the more familiar 6-sided die and replacing 4,5,6 with 1,2,3 on its faces.
The percentile die, which is a golf-ball sized polyhedron with 100 faces, is also quite iconic, though rarely used in practice due to being ridiculous. Most people just roll two 10-sided dice, instead.
You would be amazed at what tabletop gamers do and do not consider “overkill” :-)
EDIT: In the interests of full disclosure, I am a tabletop gamer, and yet I do consider crypto-quality random numbers to be overkill, but I may be in the minority on this.
Here’s a candidate for a question to illustrate a couple of related biases:
Given the following two dice roll records:
1: HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHH
2: THTTHTHHTHTTHHHTTHTHTTHHTHHTTTH
Which of the following is true:
A) 1 is more probable than 2.
B) 2 is more probable than 1.
C) Both are equally probable.
Now, I predict that there will be at least 1 “normal” person who answers C.
“Unbelievable,” you say?
Stay tuned!
I will make a stronger prediction: If this question were posed to 1000 randomly selected, well-dressed, Nordic-looking people found purposely walking the downtown sidewalks during daytime in a large American city (with luck, eliminating the possibility that I cheat by selecting 1000 people from insane asylums or from people who know no English), I predict that there will be at least 1 person who answers C.
Why? Because it is a well known fact that there exist, in much larger numbers than 1 in 1000, people capable, willing, and even eager to use the “toilet paper tube fallacy”. Any of such people combined with any of those who are susceptible to the “literalist fallacy” will answer C.
Let me make a stronger prediction. Even given a 4th choice, so slyly left out:
D) Beats me.
I predict that, still, at least one person will select C.
Now, list the following in order of probability:
a) That one person is a moron.
b) That one person is a computer programmer.
c) That one person is a card shark.
d) That one person believed that choice B was to be taken literally. That is, that B really (really!) means that the very first coin flip came out tails—NOT HEADS! - tails, the second heads, the third tails, and so on.
e) That one person ignored as much context around the dice roll question as he could. That is, that person pretended he was similar to a computer in seeing the world through what amounts to a toilet paper tube. Just the facts, Ma’am.
f) That one person is a card shark and a computer programmer.
g) b and c
h) d and e and f
i) All of the above.
“h”, anyone? :)
But, a thought on this question: How to avoid the conjunction fallacy?
Perhaps a better way to do so than keying on the word “and”, (which, as we all know, means “OR”, but not “OR and not AND”) is to key on the word “probability”. That is, when you see that word (or sense its meaning) as a goal, hand the question to the modern equivalent of a four-function calculator and let it grind out the numbers. To do so otherwise would be like multiplying 10821 by 11409 in your head, wouldn’t it?
I’m sorry—I suppose I’m probably missing something, but I can’t think of any other possible way to interpret this question. I agree that it is far more probable to see a sequence equally containing both heads and tails than one containing only heads, but it seems like you are asking for the relative probabilities of two highly specific sequences of the same length. Could someone please explain?
A. There is significantly greater than a 1 in 2^31 chance that the coin is significantly biased towards heads. This sequence overwhelms almost all priors of fairness, and thus we can conclude that the coin is almost certainly biased towards heads.
He’s rolling a die. As such, both “possibilities” are overwhelmingly improbable, as I have never seen a die labeled with heads and tails, and I spend a lot of time around dice.
Tabletop RPGs often use the term “roll M N-sided dice”, or “MdN” for short, to mean, “generate M high-quality random numbers between 1 and N”. The dice themselves are merely an implementation detail; they could be physical dice, or some random-number generator built into a collaborative RPG software program, etc. It’s common to refer to coins as “d2″s, because that’s the function that they serve.
Another interesting die roll that comes up quite often is “Md3”; the 3-sided die is usually implemented by taking the more familiar 6-sided die and replacing 4,5,6 with 1,2,3 on its faces.
The percentile die, which is a golf-ball sized polyhedron with 100 faces, is also quite iconic, though rarely used in practice due to being ridiculous. Most people just roll two 10-sided dice, instead.
When I hear “high-quality random numbers” I think “crypto-quality random numbers”—which certainly suffice, but are clearly overkill...
You would be amazed at what tabletop gamers do and do not consider “overkill” :-)
EDIT: In the interests of full disclosure, I am a tabletop gamer, and yet I do consider crypto-quality random numbers to be overkill, but I may be in the minority on this.
Yes, but those are typically the same people who have rituals around their dice. Which, on reflection, seems kinda contradictory...
Yes, but a d2 has the values 1, and 2, not heads and tails.
Okay, Felix, I have read your painfully detailed description of a hypothetically situation. Now I wanna know what your point is.