The proper choice between (1) certainly save 400 lives and (2) 90% probability of saving 500 lives with 10% probability of saving no lives, depends on your utility function, which depends on the circumstances. If your utility is proportional to the number of lives saved, then sure, go with (2).
On the other hand, suppose that some cataclysm has occurred, those 500 lives are all that remains of the human race, and extinction of the human race has such an extremely negative utility for you that all other considerations amount to rounding error in the utility function. Then, to a close approximation, you want the choice C that maximizes P(S | C), where S=”human race survives”.
We have
P(S | C=1) = P(S | N=400)
P(S | C=2) = 0.9 * P(S | N=500)
where N is the size of the current population. Therefore, you should choose (1) if
The proper choice between (1) certainly save 400 lives and (2) 90% probability of saving 500 lives with 10% probability of saving no lives, depends on your utility function, which depends on the circumstances. If your utility is proportional to the number of lives saved, then sure, go with (2).
On the other hand, suppose that some cataclysm has occurred, those 500 lives are all that remains of the human race, and extinction of the human race has such an extremely negative utility for you that all other considerations amount to rounding error in the utility function. Then, to a close approximation, you want the choice C that maximizes P(S | C), where S=”human race survives”.
We have
P(S | C=1) = P(S | N=400)
P(S | C=2) = 0.9 * P(S | N=500)
where N is the size of the current population. Therefore, you should choose (1) if
P(S | N=400) / P(S | N=500) > 0.9.