Answer:  We add the two probabilities of successes

2/5 + 3/10 in red above.

                            2/5 + 3/10 = 4/10 + 3/10 = 7/10

Or we could subtract the probability of two wrong keys 3/10 from 1
and get 1 - 3/10 = 7/10.

Explanation:  The upper left branch represents picking a right key
the first time with a probability of 2 right keys out of 5, or 2/5.
That led to a success (opening the door).  So we write 2/5 on the
branch that leads to that, and we do not branch from because we
open the door. 

The lower left branch represents picking a wrong key the first time 
with a probability of 3 wrong keys out of 5, or 3/5. So we write
3/5 on the lower left branch.  So we must branch again from there.
There are two possibilities for the second key we try.  We have
eliminated 1 wrong key, so we only have 4 keys to try, 2 are wrong
keys, and 2 are right keys.  So it's a probability of 2 right keys 
out of 4 for picking the right key the second time and also a 2 
wrong keys out of 4 for picking the wrong key the second time. That's
why I wrote 2/4 instead of reducing it to 1/2.

Edwin