Probability Question

Urn 1 has 2 white balls and 4 red balls.
Urn 2 has 2 white balls and 1 red ball.
a ball is then randomly picked from urn 1 and placed into urn 2
what is the probability of picking a white ball from urn 2?

i think ive done it wrong...
there is 0.33 prob that a white ball will be picked from urn 1
if the white ball then goes into urn 2, there is 0.75 prob that a white ball will be picked from urn 2

from this i then multiplied the 2 together 0.33*0.75 and got 0.2475, but this seems wrong because even if a red ball was moved into urn 2, there would be a 0.5 prob that a white was picked from it.

can someone help me find out where ive gone wrong.
You are not asked to pick a red ball from urn 2.

Are you familiar with conditional probability?

Try it your self with bits of paper in a bowl.

Bowl 1 containing 2 white 4 red, Bowl 2, 2 white and 1 red.
try it say 20 times and the result divided by 20.

You are forgetting that there are two paths to getting a wihite result.