Tag: Math

Categories
Ideas

Prospect Theory in Real Life OR How Losing Feels Bad More than Winning Feels Good

I’m going to do a magic trick with a number. I’m going to take a number 1700 and by doing nothing more than raising and lowering it, I’m to show how the interpretation of the number can dramatically change.  Let’s see how that can happen and then I’ll explain how that works.

When my wife was pregnant with our second son, we had a test for Downs Syndrome. This test had three parts:

  1. A “Nucal” sonogram that measured some key ratios. This was the most important test and sets the baseline.
  2. A blood test that measured blood proteins in the mother.
  3. A test of “soft markers” that refined the initial estimates based on other sonogram features.

So we had the initial test. The chance of an issue was 1 in 1700.

“Is that good?” We asked the doctor. “It sounds good to us.”

“Well, in order to be certain, you’d need to have an amniocentesis which has a 1 in 400 chance of serious problems,” said the doctor.

So 1 in 1700 is pretty darn good. Then we got the blood test back. The numbers were even better. Our chances now were 1 in 6800. That was 4 times better than we’d had before!

So we’d finished 2 or the 3 tests. Then, things got tough. We went in for a sonogram and the technician stopped at one point and said, “I need to get the doctor.” That’s never a good sign.

When the doctor came back he said, “Well, your child had 2 soft markers for Downs.”

“What does that mean?” we asked.

“Well, it means that your child has a higher chance of having Downs Syndrome. Maybe you should see a genetic counselor,” he said.

“Before we go down that route, how does this really alter our chances?” we asked.

“Well, we’re not really sure. One soft marker could double the chance of having Downs Syndrome. So 2 soft markers might increase the chance by as much as 4 times but it’s probably less than that,” he said.

“So you’re saying our chances are back to 1 in 1700.”

“Yes.”

See. Magic.

How did this happen? Behavioral Economics has an answer. In contrast to typical economic theory, Behavioral Economics looks at situations and sees how people really react — not how they would react in theory. The situation above is an example of Prospect Theory — the finding that losing something causes about twice as much pain as the pleasure you get from gaining something. So gaining and then losing the same amount still feels like a net loss.

Categories
Kids

The Liars Paradox OR Today Is Not Opposite Day

Today my son Blake started telling me that “Today is Opposite Day!” and then said things like “I love doing my homework. Just kidding. It’s opposite day!”

I told him that he couldn’t possibly be telling me that today is opposite day. If it were opposite day and he was telling me it was opposite day, then it wouldn’t be opposite day. And if it’s not opposite day and he told me that it was opposite day, it would be opposite day. It’s a cycle that never ends. Formally the sentence, “This is opposite day” is neither true nor false and therefore is undefined.

The Infinite Circle of Opposite Day

This, of course, prompted his friend Gabe to try to explain it all to me. “It’s complicated,” he said, “you see, if we say it’s opposite day then we would say that it’s not opposite day to mean that it really is opposite day.” But I didn’t find this line of argument compelling.

Blake tried a different tack, “We can say that Wednesday is opposite day.”

“Yes,” I said, “but you can’t say that on Wednesday.”

This is an ancient logical paradox called the Liar’s Paradox which often takes the form of “I am a liar” or “This sentence is false.” Because the sentence is self-referential and negative.

I figure it’s never to early to teach the kids about logic and paradox. It also makes Blake be more specific about opposite day. The inherent problem with opposite day is that kids randomly choose which items are opposite and which are not (e.g., the sentence “It’s opposite day” is not negated). Now he needs to say “If it were opposite day, I’d say that I love doing my homework.”

Categories
Fun Stuff

How Numbers Work In The Real World

I really like numbers. I understand that not everyone likes numbers as much as I do and that’s OK. I still like you.

Radiolab did an awesome podcast on Numbers a few years ago. There’s a lot of great stuff in there and it’s accessible even to people that are afraid of math. One of the most interesting bits is how our natural sense of numbers is different from formal math. Come to think of it, that may be why some people have a problem with math.

The core idea is something they refer to as natural counting. We’ve all been conditioned by math classes to think of counting as 1,2,3, etc. However, in the natural state, people act differently. On the show, they told of an experiment with an Amazon tribe where people do not count and don’t have numbers beyond five. They showed the tribesmen a line. On one side they placed 1 object and on the other side, they placed 9 objects. The experimenter asked, “What number is exactly between 1 and 9?” The response was “3.” The reasoning works like this:

  1. First person has 1 stick
  2. Middle person has 3 sticks (3 times as many as the first person)
  3. Last person has 9 sticks (3 times as many as the second person

So the second — the one in the middle has 3 sticks.

Let’s look at another example. Imagine you’re giving out bonuses at work and you have a pool of $500,000 and you want to fairly distribute the pool. You could give everyone the same amount of money. But what if someone made $1 million and 4 people made $100,000. You could give everyone $100,000 as a bonus. But that doesn’t feel fair. To be fair you’d give everyone about the same percentage of their salary as a bonus — which in this case would be about a 36% bonus for everyone.

Another non-intuitive concept on Radiolab is Benford’s law.  Benford’s law says that in the real world, you’ll see the number 1 appear many times more as the first digit than the number 9. This happens for naturally occurring phenomena like money in your bank account, size of countries or views on YouTube.

Radiolab has a great story on how this is used in the real world. Say someone was trying to commit a fraud on tax returns. They would be trying to create “random” numbers by having the first digit of each number evenly distributed (equal numbers of 1,2,3, etc.) But Benford’s law says that you should have more 1’s as the first digit than 9’s. So if numbers end up looking more random than they would be in the real world, that’s a sure sign of fraud.

So why does this happen? For the same reason that we saw above, things like to grow by percentages rather than units. Think about the bonus example. Say things are growing by 10%. If you start with the number 1 the next numbers are 1.1, 1.21, 1.33. 1.46, 1.61, 1.77, 1.94. That’s 8 numbers that start with a 1. For 9, you have 9 and 9.9 so that’s only 2 numbers that start with a 9 (and then you’re back to another number that starts with a 1).

For a more detailed explanation of Benford’s law including a pretty technical mathematical explanation, check out Singing Banana.

So why is this important? Even though we think about the world in 1, 2, 3’s, it’s actually more about the changes from where we are now. It’s 10%, another 10%, another 10%, etc. By understanding the world in these terms it’s more intuitive and more useful.