[BruceZ]

So my probability of making my flush is 9:47. Or I am a 38:9 underdog to catch my hand.

Yes and yes. Both are correct as the first expresses probability, and the second gives the odds. Two equivalent ways to say the same thing, though it would be more normal to write 9/47 for the probability.


Do I need 47:9 (or 5.2:1) odds to call or 38:9 (or 4.2:1)?

38:9 or 4.2:1.


My mind is stuck thinking that I need 47:9 and here's why: If you have X good cards of all possible cards Y. Then your chances of making your hand are X/Y. So your chances of not making your hands are Y/X. In this case, I'll make my hand 1:5.2 times and won't make it 5.2:1 times, so I need at least 5.2:1 (including my bet) from the pot to make it a good call. Not 4.2:1.
Am I right?

No.

Unlike probability, odds against are expressed as losers : winners. Pot odds are expressed as bets in pot : bets to call. You are a 38:9 underdog, so you need pot odds of 38:9 or 4.2:1. Out of 5.2 times, you will lose 1 bet 4.2 times, and win 4.2 bets 1 time.

If your probability of making your hands is X/Y, your probability of not making it is 1 - X/Y, not Y/X. Y/X is something > 1, and probability is always < 1.