As long as you impose some restriction on how the device affects your mortality, this problem can be worked out. Let T be the future lifetime of an individual. Let t be some specified time in the future. Let X be the random event that kills you. If you assume the random event affects you independently of greater mortality forces in the population, i.e. the random event doesn’t affect other factors contributing to your mortality (which is to say, the event randomly kills you according to some probability and doesn’t kill via increasing your probability of getting cancer or AIDS or something like that) then Prob(T<t and X) = Prob(T<t)*Prob(X), meaning the probability of dying from X and dying before a certain age t is reached is simply the product of the probability of dying before some age t and the probability of dying from X, which is intuitive enough. Likewise, if you go ahead and assume the device doesn’t affect the greater mortality forces governing your life and you let Y represent the probability of Y killing you, then Prob(T<t and Y and X) = Prob(Y)*Prob(T<t)*Prob(X)

Now, in order to decide whether using Y is optimal, the conditional expectations given X has occurred must be computed (the formulas I won’t describe here; they essentially give the answer to the question “what is the expected life time given that X is going to happen at some point in this person’s life?”.)

E(T given X) = (Sum over k of Prob(T=k)*k) * Prob(X)/ Prob(X) = E(T)

Which is to say, given X is going to happen at some point, the expectation of the future lifetime is simply the same as a normal life. The event occurring doesn’t affect the future expectation since it happens randomly and independent of other mortality concerns.

The other conditional expectation we need is,

E( T and Y given X ) = (Prob(Y|X)*Prob(X)/Prob(X))*(Sum over k of Prob(T=k)*k )

We want the second expectation to be greater than the first expectation, i.e. given the device is being used, the expected future lifetime is greater than had the device not been used.

So, the actual answer, instead of Prob(Y)>Prob(X): it is only optimal to use the device if Prob(Y and X)>Prob(X). That is, if the probability of the killing you AND the random event occurring is greater than the probability of the random event occurring, then it makes sense to use the device.

The question then arises how to compute Prob(Y and X). Are Y and X independent? I would not think so, since anyone who is predisposed to X would seek out the device. This however violates the assumptions X is purely random. If individuals are predisposed to X, then they probably know that and as a result are more likely to seek out the device. However, such violations do not seriously hinder the conclusion, if Prob(X and Y)>Prob(X), because it’s possible to put some boundaries on Prob(X and Y). Either they are independent, in which case, this implies Prob(Y) > or = 1 in order for it to make sense to use the device, i.e. a hundred percent success rate is need to justify the device if X and Y are independent, or they aren’t, in which case, using a common formula for Prob(X and Y) = Prob(X or Y) - Prob(X) - Prob(Y), which implies using the device is optimal if,

Prob(X or Y) > or = Prob(Y).

But the probability of X or Y is subset of the probability of Y, so the above statement can only be true when prob. of X is a superset of Y, i.e. all instances of Y are contained in X, which has no practical meaning since it would be impossible for all people who are affected by X to get access to Y without them having prior knowledge of their predisposition.

Therefore, the only way it makes sense to use the device is if it has a hundred percent success rate. This might seem counter intuitive, but it can be traced back to one assumption: The event X happens randomly. If X were to happen to happen to only a specific set of people, this would alter the future lifetime distribution for a selected person, complicating the expectation calculations.

In general, If you had detailed data on how X affects future mortality and furthermore how Y affects future mortality given X, then you could calculate the conditional expectations and compare them. If the expectation of T given Y and X is greater than the expectation of T given X, then it makes sense to use the device.