In this post we're gonna introduce one of the most widely-used results in hold 'em: the odds chart.

# Odds Chart

Before showing the odds chart, we first give the mathematical definition of odds. Here we're not focusing on winning a hand, but instead our intended issue is whether we can make an expected hand with the forthcoming unshown card(s). We call the probability of doing that as the improving probability, and define its corresponding odds as

$\text{odds} = \frac{1}{\text{improve}\%} - 1$

which means we can bet every $\1$ against any pot larger or equal to this amount.

Now we try to calculate these probabilities and odds. We here only consider one card to expect and one/two community cards to unveil, namely odds on the river or turn. For example, when we're expecting any of 8 cards on the turn to make a straight, then the improving probability in this case would be

$\text{improve}\% = 1 - \frac{45 + 2 - 8}{45 + 2}\times \frac{45 + 1 - 8}{45 + 1} = \frac{340}{1081} \approx 31.45\%$

which means we have $31.45\%$ chance to make it and the odds, therefore, is $1/31.45\% - 1 = 2.2$, which means we can bet at most $\1$ against each $\2.2$ pot. More generally, let $\#\text{n.s.}$ denote the number of community cards not shown yet, then we have

$\text{improve}\% = 1 - \prod_{i=1}^{\#\text{n.s.}} \frac{45 + i - \text{outs}}{45 + i}.$

Below is the table of improving probabilities and corresponding odds w.r.t. different outs.

Outs Improve% (River) Odds (River) Improve% (Turn) Odds (Turn)
1 2.17% 45 4.26% 22
2 4.35% 22 8.42% 11
3 6.52% 14 12.49% 7
4 8.70% 11 16.47% 5.1
5 10.87% 8.2 20.35% 3.9
6 13.04% 6.7 24.14% 3.1
7 15.22% 5.6 27.84% 2.6
8 17.39% 4.8 31.45% 2.2
9 19.57% 4.1 34.97% 1.9
10 21.74% 3.6 38.39% 1.6
11 23.91% 3.2 41.72% 1.4
12 26.09% 2.8 44.96% 1.2
13 28.26% 2.5 48.10% 1.1
14 30.43% 2.3 51.16% 0.95
15 32.61% 2.1 54.12% 0.85
16 34.78% 1.9 56.98% 0.75
17 36.96% 1.7 59.76% 0.67

# The 2/4 Times Rule

Multiplying the number of outs by two or four gives a reasonable approximation to the improve% (river) or improve% (turn) respectively, in the above table. This is a famous (yet quite rough) approximation among hold 'em gamers. The rule is a direct corollary from the abovementioned formula, as when $\#\text{n.s.}=1$,

$\text{improve}\% = 1 - \frac{46 - \text{outs}}{46} = \frac{\text{outs}}{46} \approx (2\text{outs}) \%$

and when $\#\text{n.s.}=2$,

\begin{align*} \text{improve}\% &= 1 - \frac{46 - \text{outs}}{46}\times \frac{47-\text{outs}}{47} \\ &= \frac{93}{2162}\text{outs} - \frac{1}{2162}\text{outs}^2 \approx (4\text{outs}) \%. \end{align*}