 If \(Q(H_{1}\vee H_{2}) \lt Q(H_{1}) + Q(H_{2})\), then the bookie will offer the agent the bet that pays $1 if \(H_{1}\) and 0 otherwise for \(Q(H_{1})\) and the bet that pays $1 if \(H_{2}\) is true and 0 otherwise for \(Q(H_{2})\). The bookie then buys the bet that will pay him $1, if \((H_{1}\vee H_{2})\) is true and 0 otherwise, for the price of \(Q(H_{1}\vee H_{2})\). The possible payoffs to the agent are summed up in the following table: