GRADE LEVEL: 8
The goal of this lesson is to develop the student’s intuition for inductive inference. There is some use of probability as a concept.
This lesson assumes that the concept of induction. Specifically, we define induction as a process that infers generalities from data. The generalities are theories, and the result of this process is a change in our confidence in each theory.
A 6-sided die and a 20-sided die
A screen made out of folded card so that dice can be chosen and rolled without showing the class
- Learn 3 of the most important principles of Bayesian inductive inference
- Learn that this method is universal, and that whenever we learn anything from experience we are putting these principles into practice.
When we reason inductively, we are inferring general rules from data. When we make this kind of inference, we do so by identifying which one of our possible theories best explains the data.
You will recall from the previous lesson that theories are models that make predictions. Experience and data allow us to decide which of our theories is most likely to be correct when the theories make different predictions about the frequency of an event.
In this lesson, we will play a game with dice that illustrates how this process works. In this game, we are trying to decide which of two theories is most likely to be correct. (In real world problems, we often have to start out by devising theories from scratch as new hypotheses.)
In the game, one of the dice is selected at random, out of sight of the class. By rolling the randomly selected die once, then twice, then a dozen more times, you give students more and more data about which die was rolled.
By making some arithmetic computations, we can calculate the probability that the 6-sided die was the die selected.
From this experiment, three principles are illustrated.
- All things being equal, the theory that can explain more possibilities loses to a competing theory that is more restrictive, assuming the more restrictive theory also explains the data.
- After repeated trials, evidence accumulates, and we can become extremely confident in our theory.
- Extraordinary claims require extraordinary evidence. After we have established high-enough confidence in a theory, it is more likely that data points are in error than that the theory is wrong.
This dice game is a model for how we learn anything from experience. Whenever we develop theories about the world, we are playing a game very similar to this one. As an exercise, we should be able to phrase our inferences in terms of which theory corresponds to the 6-sided die and which to the 20-sided die, and be able to explain why.
(Instructor notes will be presented in italics.)
In this lesson, we will be playing a dice game that illustrates how we reason from data to generalizations. The game is simpler than real world problems, and it is designed to make the counting of possibilities very simple. However, the principles it teaches are very general. Although we do not use these mathematical methods explicitly in our daily lives, we are essentially playing the same game whenever we make correct inferences. The great value here is that knowing how this game works can help us avoid making mistakes in some tricky situations.
Our dice game uses two dice – a 6-sided die and a 20-sided die. The 6-sided die is probably familiar to everyone. If you are not into gaming, you may not have seen a 20-sided die before. The 20-sided die works the same way, but it has 20 faces instead of 6. They are numbered 1 through 20.
In this game, I am going to choose one of the two dice at random and roll the chosen die. Your objective will be to use the information I give you to estimate the probability that I chose a particular die.
Our first step is to identify the theories that will possibly explain the data. Theory number one is… that the die I selected was the 6-sided die. Theory number two is that I selected the 20-sided die.
INSTRUCTOR NOTE: Although you will claim to have made a selection at random, choose the 6-sided die.
Okay, out of sight, I have randomly selected one of the two dice.
Based on what you know so far, what is the probability that I randomly selected the 6-sided die?
Answer: 50%. The odds are 50/50.
Yes, all you know so far is that I have made a random selection. Now I will roll my selected die. Ah… I rolled a 3.
Given this new information, what is the probability that the die I selected as the 6-sided die?
Of course, it is possible I could have rolled a 3 on either die. However, while it is equally possible to roll a 3 on either die, it is not equally probable.
Some students may not update from 50%. In fact, you have given them enough information to update their odds.
We will calculate the answer…
The probability that I selected the 6-sided die is no longer 50%. It is higher. To see why, we will imagine playing this game 120 times.
In each of the 120 games, I would first randomly select a die. In each game, I am picking randomly, with a 50/50 possibility. That means we would expect, on average, for me to pick the 6-sided die 60 times, and the 20-sided die 60 times. Now let’s see how many times we would expect 3 to come up in each of the sets of 60 games. If I roll a 6-sided die 60 times, how many times would I expect to see a 3, on average?
Since a 3 would be expected to appear once every 6 rolls, on average, we would expect to see the 3 appear 10 times in 60 games where the 6-sided die was chosen.
ANSWER: 10 times.
Now, how many times would we expect to see a 3 appear in the 60 games where the 20-sided die was selected?
Since a 3 would be expected to appear once every 20 rolls, on average, we would expect to see the 3 appear 3 times in 60 games where the 20-sided die was chosen.
ANSWER: 3 times.
Okay, so that means that we would expect to see the 3 appear a total of 10 + 3 = 13 times in 60 games. Of these, 10 out of the 13 of them would be on the 6-sided die. This is equivalent to about 77%. In other words, just by telling you the result of one roll of the die, you have changed your confidence in the 6-sided die theory from 50% to 77%.
This is how we are able to learn from experience about anything. The different possible models/theories of what is going on in the world make different predictions about the frequencies with which we see the evidence. Though multiple theories might be able to account for our data, this does not make them equivalent unless all the theories expect to see the same evidence with the same frequency.
In our example, both dice are capable of accounting for a 3 appearing in our data, but the dice generate 3’s with different frequencies.
Let’s continue with our game. I will roll the same die a second time.
Roll the 6-sided die again. The following assumes you rolled a 4
I rolled a 4. Now, do you think it is more or less likely that I chose the 6-sided die?
It is more likely, and you can leave it as an exercise later to estimate exactly how much more likely it is.
Yes, it is more likely that I originally selected the 6-sided die. What if I roll this die another 13 times, and every time I roll it I get a number from 1 to 6? What will happen to your confidence in the 6-sided die theory?
You should be very confident that I originally selected the 6-sided die.
Yes, the odds of taking a 20-sided die and rolling a 1 through 6 fourteen times in a row about 1 in 70 million. So after 15 rolls, you would be extremely confident that the 6-sided die theory is correct.
What this demonstrates is the way that repeated experimental tests cam make us extremely confident that one of our theories is true.
What would happen if I rolled the same die a 16th time and told you I rolled a 17?
The intuitive (but flawed) answer to this question is that the 6-sided die theory now has a 0% confidence, and the 20-sided die theory now has a 100% confidence.
Yes, it seems like we would know for sure that I had originally randomly selected the 20-sided die, but that your bad luck gave you 15 rolls in a row that suggested the 6-sided die had been rolled.
However, the correct answer is that we are not so sure that it was the 20-sided die. Why? Because we may have an error in our data.
How could we have error in such a simple game. ALl I have to do is roll the die, read what is printed on the top face, and read out that number to you. And all you have to do is hear what I have said.
What is the probability that I might misread or misspeak a number on the die? Or that you might mishear me? If this probability is 1 in a million, it is still much larger that the 1 in 70 million probability of making 15 rolls of a 20-sided die that look like a 6-sided die.
The correct answer is that we are no longer sure what the answer is. We now need to know more about the chances of error in our experiment in order to correctly assess which of our two theories is most likely to be true.
The lesson in this is that, once a theory has become very well established, it can no longer be displaced by ordinary evidence. Extraordinary claims require extraordinary evidence.
Using frequencies, calculate the probability that the 6-sided die was selected after the first two rolls come up as a 1 and a 6. (Hint: play 7200 games)
In 7200 games, the 20-sided die would be chosen 3600 times, and the 6-sided die would be chosen 3600 times.
Of the 3600 times the 20-sided die is chosen, the first roll will come up as a 1 exactly 180 times. In those 180 games, a 6 would show up next exactly 9 times.
Of the 3600 times the 6-sided die is chosen, the first roll will come up as a 1 exactly 600 times. In those 600 games, a 6 would show up next exactly 100 times.
Thus, out of 109 times that the sequence (1,3) shows up, 100 of them would be on the 6-sided die. This is equivalent to approximately 91.7% probability.