Question

For safety reasons, four different alarm systems were installed in the vault containing the safety deposit boxes at a beverly hills bank. each of the four systems detects theft with a probability of .99 independently of the others. the bank, obviously, is interested in the probability that when a theft occurs, at least oneof the four systems will detect it. this probability is equal to:

Answer 1

The probability any one system works is 0.99
So the probability of any one system failing is 1-0.99 = 0.01, so basically a 1% chance of failure for any one system

Multiply out the value 0.01 with itself four times
0.01*0.01*0.01*0.01 = 0.000 000 01
I'm using spaces to make the number more readable
So the probability of all four systems failing is 0.00000001
Subtract this value from 1 to get
1 - 0.00000001 = 0.99999999

The answer is 0.99999999 which is what we'd expect. The probability of at least one of the systems working is very very close to 1 (aka 100%)