Explain it to me like I am 5: The 2/4 rule

Since we now know what the words “Out” and “Equity” mean, let’s explore a very simple example that uses these concepts as well as some elementary school math (our favorite kind!) to quickly and accurately estimate a player’s chance to win the hand!

Clearly, the more Outs the better, since the more ways one has to make the best hand, the better their chance to win. Of course, the real question is: how much better is "better"? Phrasing it differently, one could ask: how much extra Equity does each Out add?

It turns out the answer is pretty simple:

Each Out is worth about 2% Equity points per card

More specifically:

Each Out is worth about 2% on the Turn (ONE card to come)

Each Out is worth about 4% on the Flop (TWO cards to come)

Before we examine why this is the case, let's first understand what these rules mean, by using a simple example:

Example:

The flop is K♣T♦2♥ and the ever-aggressive Alice makes a healthy bet! Bob looks down at Q♠J♠, an open-ended straight-draw to the nuts. In other words, any A or 9 would give him the best possible holding. Since there are 4 more Aces and 4 more Nines in the deck, Bob has 8 Outs.

Because there are two more cards to come, his 8 Outs are worth about 4% Equity each, for a total of:

This means that Bob has roughly a 32% chance of winning at showdown. With that in mind, he decides to call.

The Turn is the 3♠ and the full board now reads: K♣T♦2♥ 3♠. Alice bets big again! With only one more card to come, Bob's chances are now halved. He still has those same 8 Outs as before, but they are only worth about 2% Equity points each, for a total of:

Thus, unless Alice is bluffing, Bob will only win this hand about 16% of the time. After some deliberation, Bob decided to fold. For those wondering, Alice smiled and quietly mucked her hand.

Why does this formula work?

Hopefully, it should be clear by now how simple that formula is. To approximate their Equity in the hand, all one has to do is multiply their outs by 2 or 4, depending on whether they are on the Turn or Flop. It really does not get much simpler than this! However, is this an accurate approximation? It turns out the answer is yes. Here is why:

There are about 50 cards in the deck (52 to be exact, but who's counting?). With the exception of his 2 hole cards and the 3-4 community cards, almost all of the other cards are a mystery to Bob. In other words, there are about 50 cards that Bob does not know anything about (46-47 to be exact, but again, who's counting?).

The bottom line is that if Bob assumed that the next card will come from a pile of 50 cards, he would not be making a big error in his calculations. For instance, he can safely assume that the A♠ (one of Bob's outs) is in that pile of almost 50 cards. Therefore, the chances of getting it next are 1 in 50. Luckily, 1 in 50 is the same as 2 in 100 or 2%. Of course, with two cards to come, this percentage doubles to 4%. Thus the rule!

Wait a minute!

The astute reader may object that it is impossible for Bob to know whether a card like the A♠ is still in the deck or already in the muck. Perhaps, another player was dealt the A♠, but folded earlier in the hand. This would mean that Bob has less outs and thus less chances to win the hand. This is certainly a true statement and a valid concern.

However, the opposite situation is also possible. For instance, maybe none of the other players ever held any of Bob's outs.

Moreover, with so many people folding non-outs, the rest of the deck has way less than 50 cards but still all 8 of Bob's outs!

Thus, Bob's chances of winning increase, since each out is now worth more than just 2% per card. All in all, it turns out that these opposite scenarios tend to cancel each other out in a vacuum, making our 2/4 rule a good one.

By sacrificing a bit of accuracy, we achieve enormous gains in simplicity.

This is a trade-off worth making!

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