Independent Events

#1
Hello,

Can anyone assist with this tricky problem ?

A and B are two independent events such that:

P(A)+P(B)=t, where t is a number in the close set [0,2].

What is the maximum possible value of P(A and B) (intersection).

How do I even start ?

Thanks !
 

Dason

Ambassador to the humans
#2
Let's call x = P(A) and y = P(B).

Since A and B are independent we know that \(P(A \cap B) = P(A)P(B) = xy\)

Our goal is to maximize xy with the constraint that x + y = t (in other words... y = t-x).

This reduces down to a simple calculus problem (or just knowing where the maximal value of a quadratic function occurs) once you make the correct substitution.