Answer
$P(at~least~one~of~the~four~satellites~detects~an~incoming~ballistic~missile)=0.9999$
No.
Work Step by Step
$P(detects~an~incoming~ballistic~missile)=0.9$
The event "does not detect an incoming ballistic missile" is the complement of "detects an incoming ballistic missile". So, we use the Complement Rule (see page 275):
$P(does~not~detect~an~incoming~ballistic~missile)=1-P(detects~an~incoming~ballistic~missile)=1-0.9=0.1$
The events "satellite 1 does not detect an incoming ballistic missile", "satellite 2 does not detect an incoming ballistic missile", "satellite 3 does not detect an incoming ballistic missile" and "satellite 4 does not detect an incoming ballistic missile" are independent.
Now, using the Multiplication Rule (see page 282):
$P(none~of~the~4~satellites~detects~an~incoming~ballistic~missile)=P(satellite~1~does~not~detect~an~incoming~ballistic~missile)\times P(satellite~2~does~not~detect~an~incoming~ballistic~missile)\times P(satellite~3~does~not~detect~an~incoming~ballistic~missile)\times P(satellite~4~does~not~detect~an~incoming~ballistic~missile)=0.1\times0.1\times0.1\times0.1=0.0001$
The event "at least one of the four satellites detects an incoming ballistic missile" is the complement of "none of the four satellites detects an incoming ballistic missile". So, we use the Complement Rule (see page 275):
$P(at~least~one~of~the~four~satellites~detects~an~incoming~ballistic~missile)=1-P(none~of~the~4~satellites~detects~an~incoming~ballistic~missile)=1-0.0001=0.9999$
$0.9999=99.99$%. Altough it is secure system, we are talking about weapons of mass destruction!
