Probability of an event happening

Question

Studying for a mid-term with a practice test, and there's no solution, so I want to make sure I have this right.

A fire alarm has the property that it will ring with 99.5% probability, if there is actually a fire. On any day that there is no fire, it has a 1% chance of ringing anyway (false alarm). The fire Marshall has estimated for the building that on any single day there is a 0.03% chance of having a fire.

a) What is the probability that the bell will ring, on any given day?

b) One day, the bell rings. What is the probability that there is actually a fire?

So for the first part, I did

$$P(Bell | Fire)\times P(Fire) + P(Bell | No Fire)\times P(NoFire)$$

Plugging in I get

$$0.995 \times 0.0003 + 0.01 \times .9997 = 0.0102955 \approx 1\% $$

And for part B, I used Bayes theorem

$$\frac{P(Bell | Fire)\times P(Fire)}{P(Bell | Fire)\times P(Fire) + P(Bell | No Fire)\times P(NoFire)}$$

Which gave:

$$\frac{0.995 \times 0.0003}{0.995 \times 0.0003 + 0.01 \times .9997} = 0.028993 \approx 2.9\% $$

The reason I question my work is due to how low the second number is. Did I make a mistake somewhere, or am I correct in my work?

Thanks.

Answer 1

Looks right to me though I didn't actually check all the calculations.

The low result is initially surprising, but is not uncommon in this kind of problem. It shows that even if $P(A\,|\,B)$ is very high, this does not guarantee that $P(B\,|\,A)$ is high. It is often put in the context of a medical diagnostic test for a very rare disease.

The point is that even though the fire bell is (seemingly) very accurate, the chance of a fire happening is very very small. So the actual number of "true positives" is very very small, and will be swamped by the relatively large (though still, in absolute terms, very small) number of false positives. If it helps to see this, you could paraphrase the last fraction in your answer as $$\frac{\hbox{very very small}}{\hbox{very very small}+\hbox{very small}}$$ which is going to be. . . let's just say small ;-)

And BTW. . . don't let my explanation stop you from questioning your results as you have done! It is a very good habit to get into (or stay in).

Answer 2

Your working is correct.

The reason the number is low despite the selectivity and sensitivity of the alarm is that because there are so very few actual fires then there are still many more false alarms than true alarms.