An Introduction to Probability and Inductive Logic by Hacking Ian

Author:Hacking, Ian
Language: eng
Format: epub
Publisher: Cambridge University Press
Published: 2001-07-01T16:00:00+00:00

In the future, we will usually omit the dollar signs.

THIRD THOUGHT EXPERIMENT

Suppose that neither you nor your friend knows whether event E will occur. I offer the following deal, for nothing. You can have either.

Option (E): A chance of winning (1−p)($10) if E occurs,

or

Option (~E): A chance of winning p($10) if E does not occur.

Now you have to divide the cake—choose p. Your friend then chooses either option (E) or option (~E). If in your opinion p is a fair betting rate, then neither option has an advantage.

If, after choosing p, you think that option (E) is preferable to option (~E), then you should increase p.

This argument is just a generalization of our technique for calibrating personal probabilities.

FAIR BETTING RATES

In choosing between those two options, you did not take a risk. But the same argument will apply to anyone who bets. Suppose that the stake is $10. If p is, in your opinion, the fair rate for betting on E, then you should be indifferent between:

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