I have been watching some of the WSOP-Europe. One of the announcers has the habit of saying a player "picks up more equity" when the turn card gives the player more outs.

For example, suppose Hero gets it all-in with Ts 8s vs Villain's Ah Ad on a Js 5c 2s flop (this example is purely made up and may not make any sense poker-wise).

Then the graphics on the top of the screen will show that Hero has 9 immediate outs (the 9 remaining spades). In terms of equity, Hero has around 38% equity on the flop due to back-door draws (and the possibility of losing with a flush to a full house).

Now suppose the 7c hits the turn. Hero now has 12 outs (9 remaining spades plus 3 non-spade nines). The esteemed commentator gushes that Hero has picked up more equity! Of course, the graphics on the screen show that in fact Hero's equity has fallen to 27%.

The "negative effect" of missing on the turn is greater than the "positive effect" of Hero picking up a few more additional outs.

Okay, besides pointing out the daffy commentator, what's the purpose of this thread? I thought I would try to derive the formula for how many additional outs a "missed" turn would have to give Hero for Hero's equity to actually increase.

I simplified the derivation as follows. I assumed that Hero has X outs on the flop (ignoring any back-door draws). I then assumed that Hero misses the turn but picks up an additional Z outs. Then I solved for Z in terms of X in order for Hero's equity to increase.

If I plowed through the algebra correctly, I find that:

Z > X * (44-X) / 45

Here's a table of these values.

Outs on Flop	Reqd Addl Outs on Turn for Equity to Increase
7	5.76
8	6.40
9	7.00
10	7.56
11	8.07
12	8.53
13	8.96
14	9.33
15	9.67

You can see that it requires a hell of a lot of additional outs to make up for missing on the turn. Poker experts can weigh in on this, but I would hazard a guess to say that it is a rare occurrence.

By the way, the formula above looks like it might have a "nice" explanation. I will write it again here:

Z > X * (44-X) / 45

Of course in a heads-up situation, the turn comes from a virtual deck of 45 cards and the river comes from a virtual deck of 44 cards. Remember Hero has X outs on the flop.

Does anyone have any thoughts on this formula or anything else related to this topic??

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"Picking up more equity" doesn't automatically mean - at least in my mind - that the win chance has inctreased. It just means that his equity has increased compared to a complete blank hitting the turn.

While he lost a lot of equity simply by going from flop to turn without making a hand that is ahead he also picked up some by getting more outs.

Equity is based on the situation - and in this case the situation is 'turn' (not 'flop vs. turn'). So if I were pedantic I'd call that picking up 'turn equity'.

But to be honest the commentator should have just said "he picked up more outs" and leave it at that.