Changes

In sum, it is not the case in the Othello example that "men are just as apt to be black as things in general".

In sum, it is not the case in the Othello example that "men are just as apt to be black as things in general".

Expressed in terms of probabilities:  P(''m'') = 4/7 and P(''b'') = 1/7.

Expressed in terms of probabilities:  <math>\operatorname{P}(\mathrm{m}) = \frac{4}{7}</math> and <math>\operatorname{P}(\mathrm{b}) = \frac{1}{7}.</math>

If these were independent we'd have:  P(''mb'') = 4/49.

If these were independent terms we would have:  <math>\operatorname{P}(\mathrm{m}\mathrm{b}) = \frac{4}{49}.</math>

On the contrary, P(''mb'') = P(''b'') = 1/7.

In point of fact, however, we have:  <math>\operatorname{P}(\mathrm{m}\mathrm{b}) = \operatorname{P}(\mathrm{b}) = \frac{1}{7}.</math>

Another way to see it is as follows:  P(''b''|''m'') = 1/4 while P(''b'') = 1/7.

Another way to see it is to observe that:  <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) = \frac{1}{4}</math> while <math>\operatorname{P}(\mathrm{b}) = \frac{1}{7}.</math>

===Commentary Note 11.23===

===Commentary Note 11.23===