A house alarm system is supposed to go off if a burglar tries to break in. The o

Question

A house alarm system is supposed to go off if a burglar tries to break in. The owner knows that when there is a break in, the alarm will go of 90% of the time. However, because the alarm is getting old, when there is no break in, the alarm still goes off 20% of the time. The owner knows that there have been 3 break ins in the last 10 years and has no reason to think that there has been a rise or drop in crime. o Based on the last sentence, what is the approximate probability that there is actually a break in on a given day? o On what fraction of days do we expect the alarm to go off? o Suppose the alarm goes off. What is the probability that there is a break in?

Explanation / Answer

1. Approximate probability of break in on any given day = 3/(10x365) = 0.000822

2. Probability of alarm going off = P(alarm going off when there is a break in) + P(alarm going off when there is no break in)

= 0.000822x0.9 + (1-0.000822)x0.2

= 0.2005754

3. P(break in | alarm goes off) = P(break in and alarm goes off) | P(alarm goes off)

= 0.000822x0.9/0.2005754

= 0.0037

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