1. (*) Carmen Sandiego is equally likely to be anywhere on the surface of the Earth. Since lines of latitude are longer near the equator, it’s more likely for her to be near the equator than near the poles. So if we learn that she is somewhere on the prime meridian, then it’s still more likely for her to be near the equator than near the poles. But by the symmetry of the problem, we should expect her to be anywhere on the prime meridian with uniform probability
2. (**) You have the opportunity to play the following game: Flip a coin until you get tails, and receive 2n dollars where n is the number of times you flipped the coin. Since the expected value of this game is infinite, you should be willing to pay any price to play this game. But clearly this is absurd.
3. (*) Let P be the claim “all ravens are black”. Observing a single black raven is evidence that P is true. P is equivalent to “all non-black things are non-ravens”. Therefore observing a single non-black thing that is not a raven is also evidence of P. But this is absurd.
4 (**). You have a friend to whom blue objects appear red and vice versa (and pink objects appear light blue and so on). But since your friend has always had this condition, they believe that the word “red” refers to what they experience as blue and they use that word to refer to objects you experience as red. Thus you are unaware of your friend’s condition. How do you know you don’t have this condition? Could everyone in the world have this condition? Is there no way to know, even in principle? What is going on here?
5. (*/**) There are two envelopes full of money, and you may choose and keep one. All you know is that one contains twice as much money as the other. Suppose the envelope you choose contains n dollars. The other envelope is equally likely to contain 2n dollars as n/2 dollars, for an expected value of 3n/2 . So whichever envelope you choose, you should expect the other one to be better. How can this be?
6. (**) An instructor chooses one one of two buckets at random (50% / 50% chance): one containing 5 green and 20 red balls, and one containing 5 red and 20 green balls. Then all participants draw a ball from the selected basket (while being separated in different rooms). They are proposed the following bet:
Everyone gets 50 chips if the selected bucket was the “mostly red”
Everyone loses 100 chips if the selected bucket was the “mostly green”
The game master then asks all those who have a red ball if they want to accept the bet. If everyone says “yes”, the bet is on. A person who drew a red ball should conclude that it’s much more likely that the drawing was from the “mostly red” bucket, as the likelihood of them getting a red ball is 0.8 in that case, and only 0.2 in the “green” case. SO their expected payoff is 50*0.8 – 100*0.2= 20. The bet is favorable and they accept it
But, on the other hand, in advance you should conclude that your expected payoff in such a game is
50*0.5 – 100-*0.5 = -25 and you should reject the bet
7. (***) A stranger comes up to you on the street and asks you for $5. In exchange, the stranger claims they will use magical powers to cause a fantastic number (say N) of people in a different world to be very happy and lead rewarding lives. You don’t believe the stranger has magical powers, but surely there is an N such that your subjective probability of the stranger being able to carry out their promise is greater than 1/N. Then since the stranger can give you more expected happiness and well-being than you can otherwise buy with $5, you should give the stranger the money. But this is absurd.
8. (**) A person who does not know Chinese sits alone in a room with a big book of instructions. The person receives letters written in Chinese through a slot in the wall and, following the book’s instructions, manipulates the symbols to produce responses in Chinese. The instructions do not require the person to do anything creative or intelligent, but this process results in perfectly coherent Chinese responses to the input, as if a person had written those. So responses to the letters are being written without anyone involved understanding the letter. But how can that be?
9. (*/**/***) We expect healthy grass to be green tomorrow because we have always observed it to be green in the past. Define the adjective grün to mean “green if the date is before January 1, 2020, and blue otherwise”. We have always observed living grass to be grün, so we should expect grass to always be grün . So grass will be grün in 2020
10. (**) Discovering a cure for a disease provides an infinite stream of benefits, since people get health benefits of not having the disease in every generation. Therefore we should allocate our entire health budget to research. But clearly this is absurd.
11. (**) You are blindfolded and transported to an unknown city. You do not know how many taxis there are in the city. The only thing you know is that the taxis in the city are numbered 1,2,3,…,n, for some value of n with each number occurring exactly once. You stand on the street corner and wait until you see a taxi. The taxi is numbered k. What probability should you assign that there are at least 2k taxis in the city?
A person might argue:
(I). The probability is 50%. Because if we get dropped again and again into new cities, and always put 50% on this claim, the claim will be right half the time (because the taxi will be 50% of the time in the first half of all taxis, and 50% of the time in the second half), and this is what 50% means.
Alternatively, a person might argue:
(II). The probability cannot be determined unless you have a probability distribution over city-size (or rather, over city-taxi numbers). Imagine, for example, any specific set of cities, e.g., that there is a 1/3rd chance of 100 taxis, a 1/3rd chance of 200 taxis, and a 1/3rd chance of 300 taxis. Then, given any particular k, you can determine the chance that n > 2k using standard conditional probability. And the answer you get depends on your distribution over cities. So the person claiming (I) would be incorrect, since they are claiming an answer that does not depend on the distribution over cities.
12. (**/***) Modern cosmology teaches that the world might well contain an infinite number of happy and sad people. Aggregative ethics implies that such a world contains an infinite amount of positive value and an infinite amount of negative value. You can affect only a finite amount of good or bad. In standard cardinal arithmetic, an infinite quantity is unchanged by the addition or subtraction of any finite quantity. So it appears you cannot change the value of the world.
13. (**) Imagine that there's a very, very large population of people in the world, and that there's a madman. What this madman does is, he kidnaps ten people and puts them in a room. He then throws a pair of dice. If the dice land snake-eyes, then he simply murders everyone in the room. If the dice do not land snake-eyes, then he releases everyone, then kidnaps 100 people. He now does the same thing: he rolls two dice; if they land snake-eyes then he kills everyone, and if they don't land snake-eyes, then he releases them and kidnaps 1,000 people. He keeps doing this until he gets snake-eyes, at which point he's done. So now, imagine that you've been kidnapped.
So you're in the room. Conditioned on that fact, how worried should you be? How likely is it that you're going to die? One answer is that the dice have a 1/36 chance of landing snake-eyes. A second reflection you could make is to consider, of people who enter the room, what the fraction is of people who ever get out. Let's say that it ends at 1,000. Then, 110 people get out and 1,000 die. If it ends at 10,000, then 1,110 people get out and 10,000 die. In either case, about 9/10 of the people who ever go into the room will die. Independently on when the experiment stops, if each prisoner guesses that the chance of them dying is 1/36, because most will die, most will be wrong.
