A house is guarded by two alarms

Question

I am trying to wrap my head around the following problem
A house is guarded by two alarms.

If Alarm 1 fires, p(theft) = 80%
If Alarm 2 fires, p(theft) = 70%

If both alarms fire at the same time, what is the probability of a theft?

• Assuming both alarms are independent
• Assuming there is some dependence between the alarms

It seems simple, but I just can't seem to get it for some reason. Your help is greatly appreciated.

Answer 1

What you really want to say, for example, is

$$ P(\text{theft} \mid \text{alarm } 1) = 0.8 $$

....

Answer 2

If independent, the probability that both alarms fire erroneously is the product of 20% (probability of error for the first one) and 30% (probability of error for the second one), which is 6%. You can do this thanks to the well known rule: probability that two independent events both occur = product of the two single probabilities. Therefore, the probability that at least one alarm fires appropriately (probability of theft) is 100-6=94%.

If not independent, the method to calculate the probability of theft is similar, but it is necessary to know the degree of dependence, i.e. the conditional probability (probability that one alarm fires provided that the other fires).