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:

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

Answer: 6 alarms are needed

b) This is given by:

$np$

With n = 8 alarms, So:

$8\times0.55=4.4$

Answer: 4.4 alarms