Difference between independent Normal variables- is textbook answer wrong? This question subtracts two independent random Normal variables. I don't understand the order they subtracted the variables:

Because runner A will win, B's time will be greater, so for A to win by more than 0.5 seconds I did:
D = B - A
P(D > 0.5)
However, the answer has D = A - B:

If A won the race, then A - B would give a negative number and D > 0.5 makes no sense?
 A: First, one must assume that past running times are
normally distributed and relevant to the proposed race between A and B. Second, one must assume that, when
racing each other, their running times remain independent, which does not seem likely in practice.
At first glance, one might think that B is bound to win because B has the lower mean running time for such races. However, because of variations in running times, there is a substantial probability that either runner may win.
Presumably, we could resolve really close races
with a photograph, but we are looking for decisive
wins in which the difference is more than 0.5 sec.
I suppose you are expected to assume that when A and B race each other, that the distribution of the difference in finishing times can be found
using the usual rules for subtracting
independent normal random variables.
Here is a simulation based on ten million races (under the assumptions above), with
probabilities that should be accurate to about three places.
set.seed(2022)
a = rnorm(10^7, 13.2, .9)
b = rnorm(10^7, 12.9, 1.3)
mean(a < b)                   # A wins
[1] 0.4245494
mean((a < b) & (abs(a-b)>.5)) # A decisive win
[1] 0.3061565                 # See Henry's Comment
mean(b < a)                   # B wins
[1] 0.5754506
mean((b < a) & (abs(a-b)>.5)) # B decisive win
[1] 0.449838                  # Book answer

As stated in your question, there seems to be some confusion between the
Problem and the Answer. I will leave it to you to
decide which answer is relevant.
