Chapter 41: Induction and Probability

In this chapter the author talks a lot about probability and gives examples of how there are many flawed hypothesis’s in relation to probability experiments. The author also talks about induction and confirming laws set in continuous patterns. This chapter is full of examples that are given to try and explain each concept. The first example tells us as the reader of a continuous- never ending carpet with a certain pattern on it. On this carpet are billions of tiny triangles and whenever a blue triangle is found, this blue triangle ha a red dot in one corner. The example goes on to say that after finding thousands of blue triangles, all with red dots in the corners, the experimentalists conclude all blue triangles have red dots. This conclusion is drawn without finding every blue triangle. Would you consider this to be true? How would you know for sure without completing the complete experiment? These are some of the questions that came to mind when reading this. Gardner goes on to explain how if there are no counter examples of the hypothesis, each example is a confirming instance of the law. Many more examples are given throughout the chapter about probability and drawing conclusions from these experiments. The base of all these examples given in the text explains to us as the reader that we can’t draw conclusive outcomes from certain tests when looking at the bigger picture. The best example given is the one at the end of the chapter. This example gives you the three tables, A, B and C. On all of these tables are two hats, a black hat and a grey hat. In each hat are a bunch of poker chips, either coloured or white. On the first table (table A) the black hat contains 5 coloured chips and 6 white chips. On the same table, the grey hat contains 3 coloured chips and 4 white chips. Your objective is to draw a coloured chip, which hat will you pick out of? The black hat is the better choice because the probability is higher (5/11 vs 3/7). The same is also true for choosing at table B. You would pick out of the black hat for a better chance to draw a coloured chip (6/9 vs 9/14). Now suppose you go to table C where all the chips from the black hat in table A and B are combined to get a total of 11 coloured chips and 9 white chips for the black hat on table C. The same is done for the grey hat and you now have a total of 12 coloured chips and 9 white chips. Now your choice will be different. You would pick for a coloured chip from the grey hat because its probability is greater (12/21 vs 11/20). This situation is called Simpsons Paradox by Colin R. Blyth. This is why you can’t draw conclusions until all experiments are done. This example contradicts the first example with the blue triangle and the red dot in a way because one would imagine you would pick the black hat on table C after picking it on both tables A and B, but after doing the full experiment the hypothesis is actually incorrect. So why should this be different in the case of the blue triangle with the red dot in the pattern on the carpet? This is a thought that crossed my mind after reading the chapter. 

After reading the chapter and the reference on Simpson’s paradox, I would of like to see some graphic representation illustrated in the test written by Gardner. I think it would of been a good way for visual learners to see the difference in how the paradox works. It would’ve been an easier concept to grasp if I was able to see how the averages of the individual experiments vs the combined experiments were shown on a table or a graph. On the other hand, I am happy that Gardner omitted Blyth’s boundaries for Simpson’s Reversals. I believe these boundaries would have made the example very confusing and very mathematical instead of explaining it so we as non-mathematicians could understand it. 

Blyth notes that from a mathematical standpoint, subject to the conditions

P(A/B&C) ≥ δ . P(A/~B&C)

P(A/B&~C) ≥ δ . P(A/~B&~C)

with δ ≥ 1, it is possible to have

P(A/B) ≈ 0 and P(A/~B) ≈ 1/δ.

These are Blyth’s boundaries for Simpson’s Reversal. 

The authors of “Sex Bias in Graduate Admissions Data from Berkley” show their data in a table which makes it easier to read and view thoroughly. Much of the statistics shown in the book by Gardner are embedded into the actual text, thus making it a little more difficult to read and gather properly. Using a table example such as the one of admitted and denied male and female students like the authors of “Sex Bias in Graduate Admissions: Data from Berkeley”, it would make the chapter more organized and a better read. Having said that, I am happy that Gardner did not include the formula for putting together the graph that shows a four-cell contingency for the male and female students. I think it would have been unnecessary and confusing if it was added. 

Lastly, I am happy that Gardner omitted the triplet of confirmation concepts.

1. Classificatory- e confirms h.
2. Comparative- e confirms h more than e’ confirms h.
3. Quantitative- the degree of confirmation of h on e is u.  

I think this concept was not important to the style of the authors explanation of the theory and would have made the chapter much harder and more complex. 

Overall, the chapter gave many good examples of the theory it was explaining which made the whole concept much easier to think about and process. These examples also gave the chapter a more fun and imaginative feel to the other ones I have read, making it a more enjoyable read.