One satellite is scheduled to be launched from Cape Canavera

   One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air Force Base in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral launch goes off on schedule. If A and B are independent events with P(A) > P(B) and P(A ? B) = 0.626, P(A ? B) = 0.144, determine the values of P(A) and P(B).

Solution

P(AnB)=P(A)*P(B)=0.144
=>
P(A)=0.144/P(B)

P(AUB)=P(A)+P(B)-P(AnB)
=>
0.626 = P(A)+P(B)-0.144
=>
0.77 = P(A)+P(B)
=>
0.77 = P(B) + 0.144/P(B)
=>
0.77P(B)=P(B)^2 +0.144
=>
-P(B)^2 + 0.77P(B) -0.144 = 0
=>
P(B) = (-0.77/-2) +/- sqrt(0.77^2 -(4*0.144))/-2
=0.385 +/- 0.065
= 0.45 or 0.32

now we know P(A)>P(B) so we can calculate values of P(A) for each of these possibilities.
P(A)*P(B)=0.144, so
P(A)*0.45=0.144 => P(A)=0.32, so that can\'t be right, we must have the case
P(A)*0.32=0.144 => P(A)=0.45

So there you have it.
P(A)=0.45
P(B)=0.32