Hello. I’m currently reading Poker Maths Made Easy by Roy Rounder.

I have (2s,2c)..the flop comes (4d,2d,3d)
You’ve hit your trips. But there are a lot of draws on the board for your opponents. There’s a flush draw, straight draw, and possible straight-flush draw. All of these hands BEAT yours.
Everyone checks to you. You lead out with a medium bet and get two callers. If someone already made their flush or straight they would probably raise... so you’re putting both opponents on draws.

A flush draw at this point has nine outs and a 35% chance of completing. A straight draw has six outs if the player is holding a five... and only three outs if he’s holding the Ace.
Remember... a normal open-ended straight draw would have EIGHT outs instead of six. But we must “discount” the Ace of diamonds and five of diamonds, since those would complete the flush draws. (And the flush beats the straight.)
So... all in all there are fifteen cards in the deck that can beat you if one of your opponents is on the open-ended straight draw and one is on the flush draw. There’s a 54% chance that one of these hands will hit the board and complete the hand. (We’ll go over how I know these percentages a little later.)

QUESTION: How come the flush draw has nine outs instead of 10? And how come for the OESD, we must discount the Ad and 5d?
Can someone show me by example how to calculate the outs in the example above?

Thanks

If there are 3 diamonds on the board, and one comes to give someone a flush, that means that person already has a diamond. If they had 2, they would already be ahead of you with a flush. So if they have one, and 3 on the board, that is 4 accounted for - leaving 9 that could come and give them a flush.

If you are counting cards you don't want to see, there are therefore 9 diamonds, 4 aces, and 4 fives - but the ace of diamonds and the 5 of diamonds are already counted in the 9 diamonds. So we aren't really discounting them, we are simply not double counting them.

As for counting outs - this is usually something you do when you think you are behind. So, if you are behind, you need to get a full house (unless they have a straight flush, in which case you are toast). So there are 3 4s 3 3s and 1 2 making 7 cards that will improve your hand. If another card comes on the turn, you then have 3 of them added for the river. If you are the person with the flush draw, you have 9 outs - the 9 diamonds that are left - but you could get one and then have it pair on the river. If you have a straight draw, this is where you have to discount the Ad and 5d - you have 6 outs for a straight that does not give someone a flush. And you could also hit the straight and have it pair on the river, or have it be a diamond on the river.

If another card comes on the turn, you then have 3 of them added for the river..what does this mean?

Thanks

If we're currently behind to a straight and the turn is, say, a ten, there are now three additional outs (the remaining 10s) that will improve us to a boat