Would it he rational for an agent to hold the three beliefs P (A) = 0.4, P (B) = 0.3, and P (A V B) = 0.5? If so, what range of probabilities would be rational for the agent to hold for A ^ B. Make tip a table like the one in Figure, and show how it supports your argument about rationality. Then draw another version of the table where P (A V 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.

Agent I Belief Outcome for Agent 1 Agent 2 Stakes a A-b Bet -a Ab Proposition a Ab -- 0.4 -6 -6 3 2 4 -7 4 to 6 4 3 -8 0.3 3 to 7 -7 a v b 2 to 8 -(a v b) 0.8 2 -11 -1 -1 -1