Poisson Distribution: Prob of <=3 Wrong Connections in Building

[Punch]

A building has 2 independent automatc telephone exchanges A and B. The number X of wrong connections for A in anyone day is a poisson variable with parameter 0.5 and the number Y of wrong connections for B in any one day is a poisson variable with parameter 1.

Calculate in any particular day, the probability that there will be at most 3 wrong connections in the building given X≥2

I tried using P(X=2)P(Y=0)+P(X=2)P(Y=1)+P(X=3)P(Y=0) but the answer was wrong

 

[CaptainBlack]

A building has 2 independent automatc telephone exchanges A and B. The number X of wrong connections for A in anyone day is a poisson variable with parameter 0.5 and the number Y of wrong connections for B in any one day is a poisson variable with parameter 1.

Calculate in any particular day, the probability that there will be at most 3 wrong connections in the building given X≥2

I tried using P(X=2)P(Y=0)+P(X=2)P(Y=1)+P(X=3)P(Y=0) but the answer was wrong

\[P(X+Y\le 3|X\ge 2)=\frac{P(X=3)P(Y=0)+P(X=2)P(Y\le 1)}{P(X\ge 2)}\]CB