Are you smarter than a chimp?

I watched a great talk called, "How not to be ignorant about the world" by Hans Rosling. He demonstrates that people, regardless of education level, are often less knowledgeable about the world than they think. Hans jokes in his talk that monkeys randomly selecting the answers to his quiz do better than the people answering them!

To get the idea, try answering the four questions below to see how you go.

1. In 1950 there were fewer than one billion children (aged 0-14) in the world. By 2000 there were almost two billion. How many do UN experts think there will be in 2100?
    
   A. Four billion
   B. Three billion
   C. Two billion
    
2. What percentage of adults in the world today are literate – can read and write?
    
   A. 80%
   B. 60%
   C. 40%
    
3. On average, in the world as a whole today, men aged 25-34 have spent 8 years in school. How many years on average have women in the same age group spent in school?
    
   A. 7 years
   B. 5 years
   C. 3 years
    
4. In the last 20 years the proportion of the world population living in extreme poverty has...
    
   A. Almost doubled
   B. Remained more or less the same
   C. Almost halved

You can find the answers to the quiz at the end of this post along with the percentage of people who selected the correct answer. The questions were part of the Ignorance Survey that was conducted in the US in 2013.

So how did you go? If you didn't do very well then you're in good company with the vast majority of respondents! As you can see by the percentages, the respondents did worse than monkeys randomly choosing the right answers 33% of the time.

One of the issues Hans raised was whether the problem was due to people not reading and listening to the media. It turns out that after surveying the media with these questions, they fared no better than the general population. The problem was that even the media themselves didn't know basic facts about the world!

In the second part of the talk, Han's son Ola provided some tips for beating the chimps. He pointed out that the reason that we often get these questions wrong is due to:

1. Personal bias - we tend to generalize from our own experience which is not representative of the broader population
2. Teachers often teach outdated information based both on what they learnt during their schooling and on outdated text books
3. News bias - sensational and unusual events are more likely to make headlines and these are not representative of world events

These biases lead to the misconceptions that:

1. Everything is getting worse
2. The gap is increasing between rich and poor
3. People need wealth before social development can occur
4. Sharks are dangerous

Ola suggests that people's scores on the world fact survey will immediately and dramatically improve if they generalize in the opposite direction, recognizing instead that:

1. Most things improve
2. Global wealth can be represented as a normal bell curve with most people in the middle
3. Most people are already socially developed before they have wealth
4. Sharks are not actually very dangerous to us since they kill very few people - so recognize that fear exaggerates danger

--
Answers to the quiz (along with the percentage of respondents who selected the correct answer):

1. C (7%)
2. A (22%)
3. A (24%)
4. C (5%)

 

The Blue Eyes Puzzle (Part 2)

This follow-up post gives the solution to the Blue Eyes puzzle described here.

The answer is that the 100 blue-eyed people will leave the island on the 100th day.

The way to understand this answer is to first work through a much simpler version of the problem where there is only one-blue eyed person and one brown-eyed person on the island. Let's first try to solve it before the outsider arrives. The blue-eyed person doesn't see anyone with blue eyes, while the brown-eyed person sees one person with blue eyes. Since the blue-eyed person doesn't know whether there are any blue-eyed people on the island, he doesn't know whether he himself has blue eyes. So he will not leave the island.

However once the outsider informs the people that there is at least one blue-eyed person, the single blue-eyed person immediately knows that it must be him. Thus he leaves the island that night. Conversely, when the outsider makes her statement, the brown-eyed person doesn't yet know the color of his own eyes because he can already see someone with blue eyes. But he knows that if he does have brown eyes, then the other person will deduce their own blue eyes and leave the island that night. Or if he himself has blue eyes, then the other person will similarly not yet know the color of his eyes and will stay on the island. Once he observes the blue-eyed person leave the island that night, he then immediately knows that he himself has brown eyes.

So in this simple case, at least, the outsider's statement conveyed crucial information to the blue-eyed person that he didn't previously know and that made the difference to whether he left the island or not. Also, on observing the blue-eyed person leave the island, the brown-eyed person deduces that he himself has brown eyes.

Now consider the next simplest case where there are only two blue-eyed people and two brown-eyed people. Again, let's try to solve it before the outsider arrives. Each blue-eyed person can see one other blue-eyed person, while the brown-eyed people can see two people with blue eyes. Therefore everyone knows that there is at least one blue-eyed person on the island.

The first blue-eyed person realizes that if he himself has brown eyes then his scenario is identical to the single blue-eyed person scenario that we just analyzed. However, without the outsider's information, the other blue-eyed person cannot deduce that he has blue eyes and will therefore stay on the island. He similarly realizes that if he has blue eyes, he has no way of knowing this, so will stay on the island.

However when the outsider states that there is at least one blue-eyed person, the first blue-eyed person will now reason differently. He again realizes that if he has brown eyes then his scenario is identical to the single blue-eyed person scenario that we just analyzed. But he now knows that the other blue-eyed person would deduce that he himself has blue eyes and leave the island on the first night. So the first blue-eyed person now just has to wait until that night to see if the other blue-eyed person leaves the island. If he does, then he can deduce that he himself has brown eyes. If he does not leave, then he can deduce that he himself has blue eyes, and will leave on the second night. The other blue-eyed person reasons in the same way and also leaves on the second night. The brown-eyed person also reasons in a similar way. However, because he can see two blue-eyed people, he will wait until the second night to see what happens. When he observes the blue-eyed people leave the island on the second night, he then immediately knows that he himself has brown eyes.

So the outsider's statement, which only seemed crucial in the one blue-eyed person scenario was also crucial in the two blue-eyed people scenario. That's because it allowed each blue-eyed person to rule out the one-blue-eyed person scenario by observing what happened after the first night. It also allowed each brown-eyed person to confirm the two blue-eyed people scenario after the second night.

The same line of reasoning applies to the three blue-eyed people scenario. Without the outsider's statement, no-one will leave the island. With the outsider's statement the three blue-eyed people can rule out the two blue-eyed people scenario after the second night, deduce that they themselves have blue eyes and leave on the third night.

How do they conclude this? All three blue-eyed people hypothesize that if they themselves have brown eyes, then there are only two blue-eyed people on the island who, in turn, hypothesize that if they themselves have brown eyes, then there is only one blue-eyed person on the island. That hypothesized single blue-eyed person will deduce that he has blue eyes and leave on the first night as a result of the outsider's statement. But if no-one leaves on the first night, then that falsifies the hypothesized two blue-eyed people's hypothesis, so they would leave on the second night. But if no-one leaves on the second night, that falsifies the hypothesis of the three blue-eyed people, so they will all leave on the third night.

Conversely, the brown-eyed people will observe the three blue-eyed people leave on the third night and deduce that they themselves have brown eyes. The same style of reasoning applies to the 100 blue-eyed people scenario, with the blue-eyed people leaving on the 100th night and the remaining islanders deducing that they themselves have brown eyes.

So the outsider's statement did enable everyone to deduce their eye colors. But what new and useful information did it convey? In the single blue-eyed person scenario, it informed the blue-eyed person that someone had blue eyes. In the two blue-eyed people scenario, it informed the two blue-eyed people that the other blue-eyed person knew that someone had blue eyes. In the three blue-eyed people scenario, it informed the three blue-eyed people that the other blue-eyed people knew that the other blue-eyed people knew that someone had blue eyes. And so on.

In other words, the information that at least one person had blue eyes had become common knowledge. This meant that they all knew it, they all knew that they knew it, they all knew that they all knew that they knew it, and so on ad infinitum.

Another way to think about this puzzle is that the blue-eyed people know that they are either in a 99 or a 100 blue-eyed people scenario, but they don't know which one. Whereas the brown-eyed people know that they are either in a 100 or a 101 blue-eyed people scenario, but they also don't know which one. So the blue-eyed people will wait to see if the other blue-eyed people leave the island on the 99th night. If they do not leave, then the blue-eyed people deduce that they are in the 100 blue-eyed people scenario, and leave on the 100th night. The brown-eyed people observe the blue-eyed people leave the island on the 100th night, deduce that they are in the 100 blue-eyed people scenario, and thus know that they themselves have brown eyes. But remember that no-one will leave the island without the outsider's statement since her statement is necessary to falsify or confirm each persons' deeply-nested hypothetical about what the others know.

The key to solving this puzzle is to recognize that it has a recursive structure, observe the pattern that emerges with the simpler scenarios, and then apply that pattern to the more complex 100 blue-eyed people scenario. It's like the pattern that emerges when solving the factorial of four (or 4!). 4! = 4 * 3!, 3! = 3 * 2!, 2! = 2 * 1!, 1! = 1. Once the base case of 1! is solved, and the factorial pattern is understood, it's then easy to solve for higher-numbered factorials.

For some other expositions of the blue eyes puzzle, see xkcd and Terence Tao.

[Added Sep 24]
Bonus question: In a different version of the puzzle, the outsider comes back the day after making her public announcement, calls together all the people on the island, and makes a new public announcement:

"I'm terribly sorry, but I retract my statement from yesterday. I had a migraine that caused me to not see colors correctly so I don't actually know that I saw someone with blue eyes."

What effect (if any) will the outsider's new statement have?

The Blue Eyes Puzzle (Part 1)

I came across a fascinating logic puzzle that goes like this:

On an island, there are 100 people who have blue eyes and 100 people who have brown eyes. No-one on the island knows their own eye color. By rule, if a person on the island ever discovers they have blue eyes, that person must leave the island that night. On the island, each person knows every other person's eye color, there are no reflective surfaces, and there is no discussion of eye color. Also it is public knowledge that the islanders are perfect logicians - if a conclusion can be deduced then they will immediately do so.

One day, an outsider comes to the island, calls together all the people on the island, and makes the following public announcement:

"I can see someone with blue eyes."

Who leaves the island and on what night do they leave?

If you haven't seen this puzzle before, I encourage you to think about it before continuing.

...

All done?

Is your answer that no-one will leave the island? If so then I'm sorry to say that you are incorrect! Despite the fact that all the islanders can see many blue-eyed people, the outsider's statement does convey new information to the islanders and that information enables them to determine the color of their own eyes.

I will give the actual answer and explanation in a few days time. In the meantime, here is a suggestion to help you find the correct solution.

First try to solve a much simpler version of the puzzle. The simplest version is the scenario where there is one blue-eyed person and one brown-eyed person. Consider what the islanders can deduce before the outsider arrives. Then consider what they can deduce after the outsider makes his statement.

Now solve the next simplest version where there are two blue-eyed people and two brown-eyed people. Again consider what the islanders can deduce both before and after the outsider makes his statement. This will help you notice what the islanders learned from the outsider's statement and how it is necessary for solving the subsequent versions with three blue-eyed people and so on.

[Added Sep 22: The Blue Eyes Puzzle solution]