Probability versus odds
The conceptual difference between probability and odds is that probability is:
successful outcomes / ALL total outcomes
or
successful outcomes / (successful outcomes + failure outcomes)
Odds are expressing the same relationship in a different way:
Failure outcomes : Successful outcomes .
So you do NOT want to reduce 46:9 unless you want to find the probability of a flush, which would be written 9/46 = 1/5.222222 = 0.191489 = 19.1%.
For odds, you want to reduce
non-flush cards : flush cards
which is
all unseen cards - unseen flush cards : flush cards
which is
47 - 9 : 9 or 38:9 or 4.2222222 : 1
to hit your flush ON THE TURN CARD.
Note that the odds against are always one less than the inverse of the probability. If p is 1/5, odds are 4:1. If p is 1/23, odds are 22:1.
