Sunday, May 30, 2004

Ideas: Two types of errors

Think of a person who is being tested for a disease -- disease X. Blood from the person is drawn and examined.

Now there are two mistakes the lab can make with the blood.

1. The person can have disease X, but the lab mistakenly says that he doesn't.
2. The person does not have disease X, but the lab mistakenly says that he does.

Once we think in terms of two errors -- not one -- we have a choice to make: which error is worse?

Bottom line: People often think that you can make only one type of error, but often we can make two.

Consider another case. Suppose a man is apprehended for his supposedly commiting a crime. Now there are two mistakes that the courts can make.

1. The man is guilty, but the court thinks he is innocent and lets him go.
2. The man is innocent, but the court thinks he is guilty and puts him in jail.

Again, we have to ask ourselves which mistake is the worse mistake to make? In the American judicial system, it is worse to make the second error. In other words, it is worse to convict an innocent man than to let a guilty man go free.

Think of driving your car. You are coming up to an intersection and you can't see very well. But you know you have a green light, so you have the right to go through the intersection at your current speed. There are two mistakes you can make.

1. There is no one in the intersection, but you slow down because you think there is.
2. There is some one in the intersection, but you don't slow down because you think there isn't.

Which is the worse mistake to make? We would say the second mistake becuase the downside (for you) is greater.

Keep this in mind: There is luck in life, but often the "good luck" at some times is countered by the "bad luck" at other times. On net, you might have as much good luck in life as bad luck, so that over your entire lifetime there is no such thing as luck.

This means that you have to live your life according to the probabilities of certain things happening and by looking at which types of mistakes are worse to make.

What lessons of life do we derive from all of this?

1. You will have some good luck in life.
2. You will probably also have some bad luck in life.
3. On net, you shouldn't count on luck getting you what you want. (There is an old saying: Chance (or Luck) favors the mind that is prepared. What this really means is that preparation, hard work, and thinking about things usually gets you more of what you want in life than luck. Luck won't get you a good grade in math, but hard work will.
4. You will make mistakes in life. This is because sometimes it is a matter of what kind of mistake you make instead of whether or not you will make a mistake.
5. You have to live by the probabilities of things.
6. It is always good to think in terms of which (of the two mistakes you can make) is worse to make. You want to choose the less worse mistake to make.

As you may know, the mistakes or errors we are talking about here have names in statistics. There is a Type I mistake or error (sometimes called an alpha error) and a Type II error (sometimes called a beta error).

These errors always refer to what is called the null hypothesis in statistics.

There are two hypotheses in statistics: the null and the alternative.

The altenative hypothesis -- HA -- is the hypothesis you think is true. For example, the police arrest a man because they think he is guilty of robbing the bank.

The null hypothesis -- Ho -- is the opposite of the alternative hypothesis. In this example, the Ho is that the man is innocent of robbing the bank.

Now with respect to the null hypothesis (and usually the null is the focus of attention) you can either reject it or accept it.

If you reject the null -- in other words you reject that the man is innocent -- and the man really is innocent, then you are making a Type I error. In other words, you are saying something is false that is really true.

Type I error = false/true = something is false that you are saying is true.

If you accept the null -- in other words, you accept that the man is innocent -- and the man is really guilty, then you are making a Type II error. In other words, you are saying something is true that is false.

Type II error = true/false = something is true that you are saying is false.

Which error is worse to make? Since a Type II error here means that the man is innocent but you are saying he is guilty, and the Type I error here means that the man is guilty, but you are saying he is innocent, then probably the Type II error is worse.

Are you guaranteed of making a type I or II error? If the man is guilty and you say he is guilty, you are not making a mistake. Or if the man is innocent and you say he is innocent, you are not making a mistake. It's just that you could make a type I or II error and you want to make the one that is less bad to make.

By the way, the government thinks this way when testing new medicines. Say it is testing X new medicine.

Now here are two options:

1. The medicine is effective, but you think it is not.
2. The medicine is not effective (actually it will make people sicker), but you think it is effective.

Which mistake is worse to make? Probably the second mistake.

By the way, overtime people have seemed to figure out which of the two types of mistakes are worse to make. This is where rules come in to play. Often people say "if you can't understand a rule, you shouldn't have to obey it." Problem is that some of our rules have come about through experience.

We have a family rule to put on your seatbelt when you get into a car. Now think of this for a minute. When you are in a car, you will either:

1. end up needing your seatbelt (because you are in an accident, or
2. not need your seatbelt (because you are not in an accident).

Now which is worse?

1. Putting on your seatbelt when you don't need it, or
2. Not putting on your seatbelt when you do need it?

#2 is worse. So you make a rule: When I get into a car, I will always put on my seatbelt. After that, you don't think about things anymore. You just follow the rule. The rule has contained in it a lot of thought about type I and II errors. But you don't go through the thinking each time you get into the car, you just follow the rule.

Along comes a person and says to discard the rule because he says there is such a small probability of needing it and you ought to play by the probabilities. Yes, but sometimes the downside of a small probability is very far down. In other words, there may be a small chance that you are wrong, but if you are, you are hurt very badly.

Many rules we follow in life simply prevent reduce the chances of our making errors with big downsides.

Here are a few:

1. Never steal something even if you can get away with it.
2. Never get into a car with a driver that has been drinking.
2. Never take drugs.
4. Never drive when you are sleepy.
5. Never have unprotected sex.

Can you think of other rules that contain much wisdom in them?


Now think of the war in Iraq in terms of our two types of errors.

Some say this: We should have gone into Iraq because they might have had WMDs.

But some say this: We should not go into Iraq because they don't have WMDs.

Two options.

1. They have WMDs but we think they don't.
2. They don't have WMDs but we think they have.

Which is the worse error? It might be #1. So going into Iraq might be the course that is least likely to end up with the biggest downside.

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