Thanks for reading James! It’s a good question, let me get to it.
It’s probably easier to see what’s going on if we set some concrete numbers down. So let’s say n is ten, and the states of nature are decided by rolling a six-sided die. The state with probability p (= 2⁄6) is where the die rolls 1 or 2, and the state with probability q (= 1⁄6) is where the die rolls 3. The last state with probability 1 - p—q (= 1⁄2) is where the die rolls anything else, so 4-6.
The table’s then supposed to mean that on A, you save 10 lives if the die rolls 1 or 2, you also save 10 lives if the die rolls 3, and you save nobody if it rolls 4-6. Or, putting it another way, you save 10 lives if the die rolls between 1 and 3 (with probability 1⁄6 + 2⁄6 = 1⁄2) and save nobody otherwise.
I think something that maybe wasn’t clear is that the probabilities in the tables are supposed to be attached to mutually exclusive events. That is, if you rolled a 1, you can’t also have rolled a 3. So there’s no way of saving 10 + 10 lives, because if you save 10 lives in one way (by rolling a 1), that means you didn’t save 10 lives in another way (by rolling a 3).
Thanks for reading James! It’s a good question, let me get to it.
It’s probably easier to see what’s going on if we set some concrete numbers down. So let’s say n is ten, and the states of nature are decided by rolling a six-sided die. The state with probability p (= 2⁄6) is where the die rolls 1 or 2, and the state with probability q (= 1⁄6) is where the die rolls 3. The last state with probability 1 - p—q (= 1⁄2) is where the die rolls anything else, so 4-6.
The table’s then supposed to mean that on A, you save 10 lives if the die rolls 1 or 2, you also save 10 lives if the die rolls 3, and you save nobody if it rolls 4-6. Or, putting it another way, you save 10 lives if the die rolls between 1 and 3 (with probability 1⁄6 + 2⁄6 = 1⁄2) and save nobody otherwise.
I think something that maybe wasn’t clear is that the probabilities in the tables are supposed to be attached to mutually exclusive events. That is, if you rolled a 1, you can’t also have rolled a 3. So there’s no way of saving 10 + 10 lives, because if you save 10 lives in one way (by rolling a 1), that means you didn’t save 10 lives in another way (by rolling a 3).