Would it be rational for an agent to hold the three beliefs
$P(A) = 0.4$, $P(B) = 0.3$, and
$P(A \lor B) = 0.5$? If so, what range of probabilities would
be rational for the agent to hold for $A \land B$? Make up a table like
the one in Figure de-finetti-table, and show how it
supports your argument about rationality. Then draw another version of
the table where $P(A \lor B)
= 0.7$. Explain why it is rational to have this probability,
even though the table shows one case that is a loss and three that just
break even. (Hint: what is Agent 1 committed to about the
probability of each of the four cases, especially the case that is a
loss?)
Would it be rational for an agent to hold the three beliefs $P(A) = 0.4$, $P(B) = 0.3$, and $P(A \lor B) = 0.5$? If so, what range of probabilities would be rational for the agent to hold for $A \land B$? Make up a table like the one in Figure de-finetti-table, and show how it supports your argument about rationality. Then draw another version of the table where $P(A \lor B) = 0.7$. Explain why it is rational to have this probability, even though the table shows one case that is a loss and three that just break even. (Hint: what is Agent 1 committed to about the probability of each of the four cases, especially the case that is a loss?)