2 people play a tennis match. Probability of winning?

Question

Players A and B play a tennis match that consists of 5 SETS. The probability of A winning the first set is 1/2.

If he wins this set, his probability of winning the next set remains 1/2. If he loses, his probability of winning the next set becomes 1/4.

If he wins now, the probability of winning the next one goes back up to 1/2, otherwise, it stays at 1/4.

What is the probability that A wins the MATCH?

My attempt: I considered every possible configuration, like (WWW, WWLW, WLLWW....) and came to the answer 5/16, which is CORRECT.

I have two doubts:

1. Is there a more elegant solution to this problem?
2. Surprisingly, if we were to just find the possible 5 letter permutations containing W and L, the ones containing 3 Ws are simply ⁵C₃ and the total number of permutations are 2⁵.

Hence the probability of a random permutation to have 3 Ws is ⁵C₃ / 2⁵.

Which is also 5/16! Does this have anything to do with anything?

Note: This problem is not a duplicate, since whether the player wins/ loses, affects the probability of winning the next set.