In an archive there was a 200 meter long magnetic tape. On one side of the tape there was a recording that takes up 54 meters of the tape and on the other side, there was a recording that takes up 13 meters of the tape. The starting point of the recording is unknown. Due to some breakage, a 27 meter long part of the tape had to be cut out, starting from the 71th meter from the start of the tape. What's the probability, that both of the recordings were left intact?
The answer I get is 0.40058, although I'm not sure if my way of solving this is correct. I got it by adding all the probabilities that this could happen: $$ \frac{54}{71}\cdot\frac{13}{71} +\frac{54}{102}\cdot\frac{13}{71} +\frac{54}{71}\cdot\frac{13}{102} +\frac{54}{102}\cdot\frac{13}{102} =0.40058 $$ Is there a better way to solve this?