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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 02:58, 18 April 2009
, 02:58, 18 April 2009→Commentary Note 11.21
Stated in terms of the conditional probability:
Stated in terms of the conditional probability:
{| align="center" cellspacing="6" width="90%"
| <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b})</math>
|}
From the definition of conditional probability:
From the definition of conditional probability:
{| align="center" cellspacing="6" width="90%"
| <math>\operatorname{P}(\mathrm{b}|\mathrm{m}) ~=~ \frac{\operatorname{P}(\mathrm{b}\mathrm{m})}{\operatorname{P}(\mathrm{m})}</math>
|}
Equivalently:
Equivalently:
{| align="center" cellspacing="6" width="90%"
| <math>\operatorname{P}(\mathrm{b}\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b}|\mathrm{m})\operatorname{P}(\mathrm{m})</math>
|}
Thus we may derive the equivalent statement:
Thus we may derive the equivalent statement:
{| align="center" cellspacing="6" width="90%"
| <math>\operatorname{P}(\mathrm{b}\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b}|\mathrm{m})\operatorname{P}(\mathrm{m}) ~=~ \operatorname{P}(\mathrm{b})\operatorname{P}(\mathrm{m})</math>
|}
And this, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man.
And this, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man.