Every single decision you make at the poker tables boils down to **risk** and **reward**. It might not always seem that way, especially if you’re **clicking buttons**, but the fundamentals behind every choice do come down to this. How much do we stand to **lose** by making this play, and how much do we stand to **gain**? How often do we think each outcome will occur?

There is a very simple bit of mathematics which is key to answering these common questions in game, which can be used either in **calling** spots or **raising** spots, especially with regard to **bluffing** opportunities.

Alongside a basic understanding of poker **odds** and outcomes, and a good ability to put your opponents on an appropriate **range**, this is really all you will need in the way of maths to start understanding and beating the game, and we’ll run over the details below.

**The Maths – Calling a Bet on the River**

Whenever you are committing chips into a **pot** in poker, there’s some basic mathematics attending your actions. Let’s give a simple example.

You get to the **river** heads-up and the **pot** is **$10**. Your opponent is first to act and bets **$10** into the **$10 pot**, making it **$20 in the middle**. You have to choose between **calling $10**, **folding** and of course **raising**.

**Let’s assume** that in this case we think our opponent is either **bluffing** or has a big hand, and that we have a mid-strength value hand.

The question remains: should we **call** or **fold**? Let’s look first at the **odds** we are being laid by our opponent. When we **call** and **lose**, we are losing **$10**, that’s the amount we are **risking**. When we **call** and **win**, we are winning **$20**, the potential **reward**. Therefore we are getting **direct odds of 2 : 1** on the **call**.

How often do we need to be correct to **break even** on our **call**? It is quite simple to turn any **ratio** into a **percentage**, we simply divide the amount on the right-hand side **(1)** by the total of both sides in the **ratio (2+1)**, so the answer is **(1 / (2 + 1)) = ⅓** or **33.33%**. If we **win** more often than this, we are making **profit** and our **call** is **+EV** (positive in **equity value**).

We can see intuitively that this makes sense, since if we were to play this situation out thousands of times, and we **lose $10** two thirds of the time and **win $20** one third of the time, overall we will **break even** and neither **lose** nor **win** money. This is known as our **break-even point**.

If we believe, therefore, that our opponent is **bluffing** more often than a third of the time, we can comfortably make the **call** in this spot. Even if we **lose** the pot **60%** of the time, we will still be making profit in the long run, if we played this spot out numerous times.

For clarity, let’s consider a couple of different sizings. How often do we need our opponent to be **bluffing** for us to make profit on our bluff-catcher **call** if he’s **bluffing** with a half-pot sizing of **$5**? In this case we are calling **$5** to **win $15**, and **the direct odds are 3 : 1**. The equation becomes **1 / (3 + 1) = ¼** or **25%**. Now we only need to be correct **25%** of the time in order to **break even** on the **call**, so if we think he is **bluffing** more often than ¼ of the time, we can make the **call**.

Bear in mind that as villains bet smaller, it is often less likely that they are **bluffing**, especially in the lower stakes. To some extent you must trust your instincts, and when a bet smells like a value play, it often is!

If our opponent on the other hand **over-pots** it and bets **$20** into the **$10** pot, we will need to **call $20** to potentially **win $30**, so **the direct odds become 2 : 3**. Now we must calculate our **break-even point** using the following maths: **2 / (3 + 2) = ⅖** or **40%**. Now we must be confident our opponent is **bluffing** with a higher frequency in order for us to make the **call**.

**The Maths – Betting as a Bluff on the River**

Let’s look at the same scenario in reverse, and imagine we are the villain who was betting full pot on this **river**. Let’s further imagine that we have nothing, and we are simply **bluffing** to try to take down the pot. How often does a **$10 bluff** into a **$10 pot** need to work in order to **break even**?

Once again the maths is very simple. In this spot we are **risking $10** for the chance to **win $10**. There’s no way to win any more than this, since we are **bluffing** and when called we will lose the **pot**. **The direct odds are 1 : 1**. In this case the maths needed to get a **break-even** percentage are even simpler, it is simply **1 / (1 + 1) = ½** or **50%**. If the **bluff** works half the time, we will **break even** on our play in the long run.

Let’s consider the **bluff** using different sizings. If we **bluff** half-pot, this would be **$5** into a **pot** of **$10**. Now we only need the **bluff** to work with the following frequency to at least break even: **1 / (2 + 1) = ⅓** or **33.33%**. Since we are **risking** less, the **bluff** needs to work less often to **break even** or to turn a profit. Bearing in mind of course smaller **bluffs** are generally prone to work less often.

If we **bluff** an over-pot sizing, say **$20** into the **$10 pot**, now we must use different inputs. We are **risking** twice as much as we stand to gain if the **bluff** works. Now the maths equates to **2 / (1 + 2) = ⅔** or **66.66%**, so we need the **bluff** to work two thirds of the time just to **break even**.

Whether we are considering a **call** vs. a **polarized range** (either **bluffing** or a very strong hand) or a **bluff** versus an opponent, the basic maths is the same, since it boils down to **risk** evaluated against **reward**. How much are we **risking**, how much is the **reward**, and how often do we think we will **win** or **lose** in the spot?

These are the fundamental questions, and the basic maths behind them is always the same: **risk divided by (risk + reward) = our break-even point. **

**Why on the River?**

We have used the examples above, situated on the **river**, for one very simple reason – no cards are left to come out on the board. This means that it is the final round of betting, and when we call or fold to that final bet, the hand is concluded. This means that we need only think about **direct odds**, the **ratio** we have given above which indicates the **ratio** between **risk** and **reward**.

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