Compound Probabilities

#1
Consider a video game where a handheld explosive device is thrown at a vehicle. Just for this example, two types of damage are considered relevant for consideration. First, a window breaks from being hit by a piece of shrapnel from the device. Second, the finish of the vehicle is burned from the conflagration. Both of these types of damage have an independent probability of occurring (Pbreak, Pburn), based on various factors such as the location of the device in 3D space when it explodes. I am calling them independent because one or the other may occur, or both may occur, or neither may occur; and the occurrence of either one has no effect on the probability of occurrence of the other. How do I calculate the probability of relevant damage occurring? That is, the probability that ANY relevant damage will occur.

My first thought is that I should sum the probabilities: Pdamage = P(break OR burn) = Pbreak + Pburn. But this can lead to a Pdamage greater than 1.0.

For a given blast location, let's say Pbreak is 0.6 and Pburn is also 0.6. What is Pdamage? Doing the above results in 1.2, which can't be right. Thanks in advance for any insight!
 

Karabiner

TS Contributor
#2
Just consider the probabilties of non-events, not the probabity of events.

Say, you have 100 explosions. You know that 40 of them result in NO break.
How many of the 40 NO breaks have also NO burn?

With kind regards

Karabiner
 
#3
To expand on Karabiner’s answer, there are four possible damage scenarios to consider, namely:
  1. Break AND Burn;
  2. Break AND (NOT Burn);
  3. (NOT Break) AND Burn; and
  4. (NOT Break) AND (NOT Burn).
Each scenario has a probability of occurring associated with it that is straightforward to evaluate, and scenarios 1. OR 2. OR 3. result in any damage, but NOT 4. (Hint: the logical operators—“AND”, “NOT”, “OR”—translate directly and uniquely to certain mathematical operations.)

This should also clarify why Karabiner’s suggested approach works.