I messed this one up! See, I tattooed ALWAYS HYPOTHESIS TEST on my forehead a few years ago, but I forgot that I can't see my own forehead, so people staring directly at me hypothesis test correctly, but I keep forgetting. Yeah. That's the reason.1. You have a weighted coin that comes up heads 66% of the time and tails 33% of the time. You flip it twenty times. Which of the following sequences do you think is most likely to exist among those twenty flips?
a) HHHHHH
b) HHTHHT
c) HTTTTH
d) TTTTTT
e) All are equally likely
Anyway, after Neike's post it should've been correct. Many of you fell for the logical error here, which is not doing the coin tosses independently. The probability of any one coin toss being heads has nothing to do with the probability any other coin toss was heads. Since the coin was more likely to come up heads than tails, the most likely sequence is all heads.
Quick demo of this - simplify it to a three coin toss. The coin's landed heads the past two times. What's it going to do now? Well, with 66% chance, it will come up heads again. So HHH is more likely than HHT, and HHH-HHH is more likely than HHT-HHT.
A completely different question would have been "Which is more likely - some combination of two tails and four heads, or six consecutive heads?" The answer THEN would have been some combination of two tails and four heads.
I've screwed up this question enough that I might still be wrong, so someone check me on this.
Well, there's no one "most probable" and "least probable" answer. But what you should have noticed were statements 2, 3, and 5. Both "Linda is a bank teller" and "Linda is active in the feminist movement" MUST both be more likely than "Linda is a bank teller and is active in the feminist movement." That is, since every time Linda is a bank teller and a feminist, Linda is also a feminist, 2 and 3 are both ALWAYS more likely than 5.2. Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Rank the following statements from most probable to least probable:
1. Linda is a teacher in an elementary school.
2. Linda is active in the feminist movement.
3. Linda is a bank teller.
4. Linda works in a bookstore and takes yoga classes.
5. Linda is a bank teller and is active in the feminist movement.
6. Linda is an insurance salesman.
This is a classic example of the Conjunction Fallacy, a logical fallacy in which people mistakenly say that "Both A and B" is more likely than "A". There are a lot of other good examples at the link.
So all of you who listed "They're all equally probable" were wrong
Serves you right.Only Ari got this one right, and that's probably because he reads the same articles I do3. Consider a person randomly chosen from the phone book named John. We investigate John's personality and find that he is quiet, bright, and likes poetry and classical music. Which of the following is MOST likely true?
a) John is a truck driver
b) John is a professor at an Ivy League school
c) Both of these are equally likely.
There are probably only a few thousand Ivy League professors in the world, but a few million truck drivers. So, even if Ivy League professors are a hundred times more likely than truck drivers to like poetry and so on, the guy's still ten times more likely to be a truck driver.This is called the Base Rate Fallacy. You'll see it again in the taxi cab problem.
Liam's answer of $50 (along with several others) seems reasonable. For all of you who said they would never press the button for any amount of money, consider that you have about a one in five hundred thousand chance of dying in an accident for every hour you're driving. If you wouldn't press the button for a billion dollars, that means that, if someone who lived a half hour away offered you a billion dollars and all you had to do was show up at his house to pick it up, you wouldn't take the offer.4. A man has a machine with a button on it. If you press the button, there is a one in five hundred thousand chance that you will die immediately; otherwise, nothing happens. He offers you some money to press the button once. What do you do? Do you refuse to press it for any amount? If not, how much money would convince you to press the button?
Or if you ever go shopping, consider that you're exchanging a small chance of death - even just by walking to the store - for however much money you'd lose by using one of those online grocery services.
I heard this example a long time ago, and I don't remember the official name for it or what study is was discovered in. If Ari or anyone else knows, please remind me.
The cab was probably green, although, as Ari snarkily pointed out, it's only a minor difference and not enough to build a court case on. This question is similar to the truck driver question in that you have to use your information to adjust from an original base rate. The calculation, which Ari obviously did, was:5. A cab was involved in a hit-and-run accident. Two cab companies serve the city: the Green, which operates 85% of the cabs, and the Blue, which operates the remaining 15%. A witness identifies the hit-and-run cab as Blue. When the court tests the reliability of the witness under circumstances similar to those on the night of the accident, he correctly identifies the color of the cab 80% of the time and misidentifies it 20% of the time. So...
a) Believe him. The cab was likely blue
b) Don't believe him. The cab was probably green.
c) The cab had an equal chance of being either blue or green.
d) There's no way to tell the probability from this information.
85% original chance of being green * only 20% chance the guy was wrong = 17% from this alone that cab was green
15% original chance of being blue * 80% chance the guy was right = 12% chance from this alone that cab was blue
But since we know the cab was either green or blue, we can find the actual chance it was green by 17%/29% = 58.6%
I did a terrible job explaining that, didn't I? No matter, go to the source: Representativeness Heuristic
Hey, pat yourself on the back. You all got this one right. There's a common fallacy of being more worried about scary, catastrophic forms of death than boring ones, even though the boring ones happen much more often. As you all figured out, saving 20% of cancer patients saves a whole lot more people than 95% of terrorism victims.6. You're a politician working on the budget. Which of these seems most worthy of funding?
a) An special team to stop terrorist attacks. Policy analysts say it will stop 95% of planned attacks on your country's soil.
b) Research for curing cancer. Scientists think it will save the lives of 20% of cancer patients.
c) Funding more policemen and detectives. This could cut the homicide rate by 66%
As it happens, the statistics suggest the cancer option ends up saving the most people, but the difference between the cancer and homicide statistics isn't big enough that you get minus points for not knowing it.
Yet another question dealing with your imminent demise. Liam was the only one to get this one right - you're most likely to die from an asteroid strike (Andreas' caveat that it's D for him because he lives in Australia has my sympathy, though).7. Which of these are you most likely to die from?
a) Terrorism!!!
b) Earthquake!!!
c) Asteroid strike!!!
d) Snakebite/spider bite!!!
e) All approximately equal!!!
Although the odds of a big asteroid hitting Earth are very low, if it does, it will kill everyone or almost everyone. If a big asteroid only hits once in a million years, but kills all seven billion people on Earth, then on average, seven thousand people a year die of asteroid strikes. In contrast, in first world countries, only about a hundred people a year die of bites from venomous animals, only about 30 a year from earthquakes, and only a few hundred from terrorism.
The actual odds of dying in an asteroid strike are disputed because we don't know exactly how often big asteroids hit. I've seen anywhere from 1/50000 to 1/6000. But everyone agrees they're rather high compared to many more common forms of death.
Ironically, once again I messed up on this question - I made exactly the same mistake I was testing you for. I took the death rate statistics from 2004. But, of course, if once every few thousand years there's a gigantic earthquake that sinks Los Angeles or Tokyo into the sea, that probably will raise the odds pretty significantly. And if once every few thousand years terrorists nuke a major city, that will probably increase them too. I should've stuck with something simpler.
But you all said snakebite, or all equal, so you're still wrong
At least we're wrong together.The most obvious answer, "d", is as usual wrong. Any word that ends with "ing" must necessarily also end with "g". There are also some words that end with "g" that don't end with "ing", so b>d. The other two were just stuff I threw in there so it wouldn't be so obvious what I was doing, but they should both be pretty rare.8. In a standard novel, which of the following would you expect to see most often?
a) Words beginning with the sequence "fij"
b) Words ending with the letter "g"
c) Words beginning with the letter "q"
d) Words ending with the sequence "ing"
This is an example of both the Conjunction FallacyConjunction Fallcy[/url], as mentioned above, and the Availability Heuristic, in which people think that if it's easier to think of an example of something, it must be more common. It's easy to think of examples of "ing" words because you can just add "ing" to the end of lots of verbs. It's harder to think of examples of "g" words because aside from the ing words, there aren't many.
Another Availability Fallacy one. Our minds find it easy to sort words by first letter, so it's easy to think of examples of words beginning with "k". It's harder to think of words with "k" as the third letter, but my source informs me there are, indeed, more.9. And which are there more of?
a) words beginning with K, like "king"
b) words with k as the third letter, like "rake"?
This was my question to see who was cheating. C, the answer that's obviously correct, in fact IS correct. If you put down anything but C, you were obviously expecting a trick question and changing your answer to something else without knowing why. Shame on you. *cough* Benkern and Maksym *cough* :P10. Counting from one to a million, which are there most of:
a) Numbers with "3" as the last digit
b) Numbers with "9" as the last digit
c) Numbers with either "2" or "5" as the last digit
d) All the same.
In the original experiment, the answer was "The three numbers are in increasing order."11. I am teaching a class, and I write upon the blackboard three numbers: 2-4-6. "I am thinking of a rule," I say, "which governs sequences of three numbers. The sequence 2-4-6, as it so happens, obeys this rule. Each of you will find, on your desk, a pile of index cards. Write down a sequence of three numbers on a card, and I'll mark it "Yes" for fits the rule, or "No" for not fitting the rule. Then you can write down another set of three numbers and ask whether it fits again, and so on. When you're confident that you know the rule, write down the rule on a card. You can test as many triplets as you like." Here's the record of one student's guesses:
4, 6, 2 No
4, 6, 8 Yes
10, 12, 14 Yes
At this point the student wrote down his guess at the rule. What do you think the rule is? Would you have wanted to test another triplet, and if so, what would it be?
This is from an experiment by Peter Wason to test for a form of Confirmation Bias. The best way to test a theory is to look for ways to prove it wrong. If you don't find any, your theory's probably right. But most people, instead, will look for ways to prove the theory right. If you thought it was increasing by 2, and wanted to test 3,5,7, you were probably looking for a way to prove your theory right.
This bias is particularly interesting because of its political implications. Read the Confirmation Bias article above for more information.
The point of this was the confidence levels. I was testing for something called the Overconfidence Bias, which says that people tend to be much more confident than the situation warrants. The article linked to above says people tend to have 60-70% confidence when they're right 50% of the time (it also contained a much better test for overconfidence that I would've used here if I'd known about it last week; take it if you want).12. The biggest city is Tokyo. The largest empire was Britain. The smallest planet is Mercury. The religions go Christianity, Islam, Hinduism, Judaism.
Take a look at your answers and see if you were overconfident in them. Maybe one of the resident math geniuses knows an exact formula to use to determine whether you were overconfident, but it's getting late for me, so here's the quick and dirty way - since there were only 4 questions, if you assigned a probability greater than 75% to any you got wrong, you were overconfident.
A fun implication of this is that if you ask people whether they're average, below average, or above average in some category - let's say as drivers - about three quarters of people will say they're above average, which is obviously false.
This was intended to test hindsight bias, aka the "I knew it all along" effect. If you know the answer to something, the answer looks obvious. There's a fun study about a flood up at Overcoming Bias' hindsight bias post.13. I lied. All of these are actually the opposite of the correct answer, except for the third, which was indeed correct.
Surprisingly, none of you had any major hindsight bias. That's probably because this was a terrible method of testing for it. Most of the decent studies in the article above required giving each condition to different groups of people, and we didn't have enough Shirerithians for that.
Or maybe you're all just geniuses.
So, why did I make you take this test? Well, aside from just really enjoying tricking people, the point was to showcase some flaws that people are very prone to making. And there are lots of other biases I forgot, and lots of others that were too hard to think of test questions for. A lot of these flaws have very, very deep effects on topics like politics, economics, religion, and society. The article on Confirmation Bias, which I again highly suggest, is a great example of those.
Even when you're taking a test with clear, concrete answers that you KNOW is full of trick questions, you're prone to making these mistakes. Even when I was WRITING this test about a subject I've studied for a long time and doing it specifically thinking about these mistakes in order to trip the rest of you up, I was prone to making these mistakes. That's not a good sign. It probably means that in our daily life, dealing with much more complex issues, we probably make many, many more invisible mistakes of the same sort.
I believe that part of the reason why people and countries keep on making bad decisions, time after time, and having the same arguments without any resolution, is because of these mistakes. That's why it's so important to learn logical procedures for decision-making. Not so that you can avoid these, because that's almost impossible, but so that you can minimize and route around them. I already posted one very weak example - the expected utility analysis of global warming. As I learn more, I'll try to post those two, and if you learn some, I want to hear about them.
Another thing I hope you've learned - just because something's a trick question doesn't mean you should always answer "They're all equal." Yvain glares at a few people
Out of interest, I wanted to figure out how wrong I was, and found out 
