Question

A large bank vault has several automatic burglar alarms. The probability is 0.55 that a single alarm will detect a burglar.(a) How many such alarms should be used to be 99% certain that a burglar trying to enter is detected by at least one alarm? (Enter the smallest number of alarms needed to be at least 99% certain.) ___alarms(b) Suppose the bank installs eight alarms. What is the expected number of alarms that will detect a burglar?___ alarms

Answer 1

a) We have:

p = 0.55

then

`1-p=1-0.55=0.45`

Suppose the number of alarms need is n, so we have:

99% = 0.99

`\begin{gathered} 1-(0.45)^n=0.99 \\ 1-(0.45)^n-1=0.99-1 \\ -(0.45)^n=-0.01 \end{gathered}`

Multiply by (-1) on both sides:

`\begin{gathered} (0.45)^n=0.01 \\ n\log (0.45)=\log (0.01) \\ n=(\log (0.01))/(\log (0.45)) \\ n=5.77\approx6 \end{gathered}`

Answer: 6 alarms are needed

b) This is given by:

`np`

With n = 8 alarms, So:

`8*0.55=4.4`

Answer: 4.4 alarms