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Revision as of 02:58, 18 April 2009
, 02:58, 18 April 2009→Commentary Note 11.21
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: P(''b''|''m'') = P(''b'')+: P(''b''|''m'') = P(''b'' & ''m'')/P(''m'')+: P(''b'' & ''m'') = P(''b''|''m'')P(''m'')+: P(''b'' & ''m'') = P(''b''|''m'')P(''m'') = P(''b'')P(''m'')+
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.
