a) We have:
p = 0.55
then
[tex]1-p=1-0.55=0.45[/tex]Suppose the number of alarms need is n, so we have:
99% = 0.99
[tex]\begin{gathered} 1-(0.45)^n=0.99 \\ 1-(0.45)^n-1=0.99-1 \\ -(0.45)^n=-0.01 \end{gathered}[/tex]Multiply by (-1) on both sides:
[tex]\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}[/tex]Answer: 6 alarms are needed
b) This is given by:
[tex]np[/tex]With n = 8 alarms, So:
[tex]8\times0.55=4.4[/tex]Answer: 4.4 alarms
