Stats ?: Device has X% chance to prevent death, Y% chance to cause death. Use it?

Consider that there is a random event which can instantly kill you (e.g. lighting strike). Based on your lifestyle, you have an X% chance of being a victim of that event.

To save you from that event, you could keep a device with you which would save your life in that event. However, at any time the device has a Y% chance of killing you.

How do you calculate the optimum percentage when you should use the device? That is, you live longer with the device than without. Is it a simple matter of when X>Y? I would think not, since I am increasing my chance of death with the device (X% from the random event and Y% from the device).

As an example, consider a smoke detector. It can save you from dying in a fire, but it could give you cancer from the radiation of the detector. However, the chance of cancer is so low compared to the chance of being in a fire that it’s clearly better to use a smoke detector. But when the odds of the event is low (lightning strike), it’s not so clear when to use a lightning strike preventer if it had a Y% chance of killing you at some other time.

If your X% and Y% are both for the same period of time, and if the device is guaranteed to save you from the random event should it occur, then yes, it is just as simple as X>Y.

But you said that the device eliminates the chance of dying from the random event, so X% goes to zero while wearing the device and your overall chance of dying is Y%.

Is this like a pharmaceutical, which for example, reduces the chance of remission of cancer from 30% to 0.05%, but has a 3% chance of accelerating heart disease? Because we can monitor each separately, and treat each one, effectively reducing the risk to .05%X/.05%Y? Or do you really meant the perfect situation you described – prevents 1 in a million occurrences with a certain low percentage of certain death. Heck, I can even apply that to yours – a device that prevents lightning strikes with a risk of cancer doesn’t have to be on when I’m home, or on my vacation in the Sahara, or in a cave, etc.

It was the perfect situation. You could remove the device when you were at lower risk if you wanted.

Common sense was telling me it was worth it when X>Y, but you never know with stats. Thanks for the answers.

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.