Exercise 13.4

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?)