Poker & Math Basics: Expected Value, Equity & Pot Odds
Expected Value (EV)
The expected value (EV) is a way to measure whether a decision you make will be profitable in the long run if you repeated the same situation many times.
The Core Idea
- Positive EV (+EV): A decision that on average wins you money over time.
- Negative EV (–EV): A decision that on average loses you money over time.
Even though you may lose in the short term, if the EV is positive, that play is correct because over many hands it will make you money.
How It Works
You calculate EV by multiplying each possible outcome by the probability of it happening, then summing those up:
$$EV = (\text{Probability of Winning}*\text{Amount Won})\\−(\text{Probability of Losing} * \text{Amount Lost})$$
Example
A fair coin has a 50% chance of landing on heads and 50% on tails. While short-term results may deviate from this (e.g., 60% heads in 100 flips), the long-run average approaches 50%. With a very large number of flips, such as one million, the proportion of heads will almost certainly fall within 49.5%–50.5%.
Now consider the following game:
- If the coin lands heads, you lose $100.
- If it lands tails, you win $200.
The expected value (EV) of the game for you is:
$$EV = (0.5 * 200) + (0.5 * -100)\\= 100 – 50\\= +50$$
Thus, your expected profit is $50 per flip, while mine is an expected loss of $50 per flip.
Example in Poker
Imagine you’re on the river, facing a $50 bet into a $100 pot.
- If you call and win, the pot is $150 + your call = $200 total. You win $150 (since your $50 call comes back to you plus the $100 in the pot).
- If you call and lose, you lose $50.
Suppose you think you’re 40% likely to win the hand if you call.
$$EV = (0.40 * 150) − (0.60 * 50)$$
$$EV=60−30=+30$$
So, the EV is +$30. That means calling is a profitable decision in the long run, even though you’ll lose 60% of the time.
✅ Key takeaway: EV isn’t about what happens this hand, it’s about what would happen if you faced the same spot 1,000 times.
Positive EV = good decision, negative EV = bad decision.
Equity & Pot Odds
Instead of crunching EV math every time, you just compare two things:
- Equity (Your Chance of Winning)
- How often you expect your hand to win if all cards were shown.
- Sometimes you estimate based on experience, sometimes you calculate using outs.
- Pot Odds (Risk vs Reward)
- This tells you the minimum % of the time you need to win for your call to be profitable.
Equity
In any poker hand, you either hold the best hand or a weaker one at every stage. Having the best hand does not guarantee winning the pot, but with knowledge of an opponent’s likely holding—or the recognition that your current hand can only win if it improves—you can estimate your chances of success.
For example, suppose you hold 8♠7♠ on a board of A♦6♣5♥K♠. If your opponent has bet both the flop and turn, your 8-high is almost certainly behind. However, a 9 or 4 on the river would give you the best hand. Out of the 46 unknown cards (52 total minus your 2 hole cards and 4 board cards), 8 improve your hand. This gives you an equity of 8/46 ≈ 17.4%.
In dollar terms, with a $100 pot, your share is worth $17.40; with a $500 pot, it is worth $87. Put differently, for every dollar added to the pot at this point, your long-term return is $0.174.
| Hand Matchup | Example | Equity Hand 1 | Equity Hand 2 |
|---|---|---|---|
| Pocket Aces vs. Pocket Kings | AA – KK | AA ~82% | KK ~18% |
| Pocket Kings vs. Pocket Queens | KK – QQ | KK ~82% | QQ ~18% |
| Pocket Pair vs. Lower Pair | JJ – 77 | Higher Pair ~80% | Lower Pair ~20% |
| Pocket Pair vs. Two Overcards | 77 – KJ | Pair ~55% | Overcards ~45% |
| Pocket Pair vs. One Overcard | 88 – J4 | Pair ~70% | Overcard ~30% |
| Pocket Pair vs. Two Undercards | JJ – 64 | Pair ~82% | Undercards ~18% |
| Pocket Pair vs. Dominated & Undercard | KK – KQ | Pair ~92% | KQ ~8% |
| Two Overcards vs. Two Undercards | KJ – 97 | Overcards ~63% | Undercards ~37% |
| Ace-King (suited) vs. Pocket Pair | AKs – 22 | AKs ~46% | Pair ~54% |
| Ace-King (offsuit) vs. Pocket Pair | AK – TT | AKo ~43% | Pair ~57% |
| Suited Connectors vs. Overcards | 87 – QT | SCs ~40% | Overcards ~60% |
| Dominated Hands | AK – AQ | AK ~73% | AQ ~27% |
| Dominated Hands | KT – QT | KT ~70% | QT ~30% |
| Two Hands with Interlocking card | KT – Q8 | KT ~62% | Q8 ~38% |
| Two Hands with Two Interlocking cards | A4 – J9 | A4 ~57% | J9 ~43% |
| Random Hand vs. Pocket Aces | XX – AA | Random ~12% | AA ~88% |
Quick way to estimate equity
The Rule of 2 and 4
A shortcut to estimate your chance of hitting a drawing hand (your equity) based on your outs:
On the flop (2 cards to come):
Multiply your outs × 4 ≈ % chance to hit by the river.
On the turn (1 card to come):
Multiply your outs × 2 ≈ % chance to hit on the river.
Example 1:
Flush Draw on the Flop
- You have 9 outs (any remaining spade).
- 9 × 4 = 36% chance to hit by the river.
(Actual probability is ~35%, so very close.)
Example 2:
Open-Ended Straight Draw on the Turn
- You have 8 outs (4 cards on each end).
- 8 × 2 = 16% chance to hit on the river.
(Actual probability is ~17%, so again very close.)
Pot Odds
Whenever you face a bet, you must decide whether the potential reward justifies the money you risk. This is measured by pot odds, defined as the ratio of the total pot (including the bet you are facing) to the cost of calling:
$$\text{Pot Odds}=\frac{\text{Call Amount}}{\text{Pot Size}+\text{Call Amount}}$$
For example, if your opponent bets $20 into a $40 pot, the total pot becomes $60.
To continue, you must invest $20 for the chance to win $60.
This gives pot odds of 60:20, or 3:1.
To convert pot odds into the minimum equity required, divide “1” by the sum of both sides of the ratio.
In this case:
$$\frac{1}{3+1} = \frac{1}{4} = 25\%$$
This means you must win at least 25% of the time for the call to break even.
| Pot Odds (Ratio) | Break-Even % (Equity Needed) |
|---|---|
| 1:1 | 50% |
| 2:1 | 33% |
| 3:1 | 25% |
| 4:1 | 20% |
| 5:1 | 16.7% |
| 6:1 | 14.3% |
| 7:1 | 12.5% |
| 8:1 | 11.1% |
| 9:1 | 10% |
| 10:1 | 9.1% |
The table above lets you quickly compare your hand equity (chance of winning) with the pot odds you’re offered. If your equity is higher than the break-even% shown, calling is profitable in the long run.
Example 1: Facing a River Bet
- Pot = $300
- Villain bets $100
- You must call $100 to win $400 total
$$\text{Pot Odds}=\frac{100}{(300+100)}\\=\frac{100}{400}=0.25 (25\%)$$
You need to win at least 25% of the time to make calling profitable.
If you think your hand is good 40% of the time, your call is +EV.
Example 2: Drawing Hand
Say you’re on the flop with a flush draw (9 outs, ~36% to hit by the river).
- Pot = $100
- Villain bets $50
- Call = $50
$$\text{Pot Odds}=\frac{50}{(100+50)}=33\%$$
Your equity ≈ 36% → higher than 33% → profitable call (+EV).
Shortcut Rule of Thumb:
If your equity > pot odds → Call is profitable (+EV).
If your equity < pot odds → Call is losing (–EV).
Why This Helps
Now, when you face a bet:
- Use Rule of 2 and 4 to estimate equity.
- Use Pot Odds formula to find breakeven %.
- Compare them → instant EV decision without math headaches.
Putting it All Together
When you face a bet, once you know both how often you’ll win and how much you stand to win, you can calculate the expected value (EV) of your decision.
For example, suppose you hold 8♠7♠ against an opponent’s A♦K♣ on a board of A-6-5-K.
Your opponent shoves $32 into a $50 pot. To decide whether to call, compare your pot odds with your chance of winning:
$$\text{Pot odds}=\frac{32}{(50+32+32)}=28\%\\ \text{Your Equity = Chance to hit your draw}=17.4\%$$
Since 28% (the price to call) is greater than 17.4% (your winning chance), this call would be unprofitable, so you should fold.
Now imagine the same shove of $32, but into a $132 pot:
$$\text{Pot odds}=\frac{32}{(132+32+32)}=16\%$$
Your equity is still 17.4%. Here, 16% < 17.4%, so the call becomes profitable — you should call.
l win about 17% of the time, but I only need to win 16% of the time to break even, so calling makes sense.
Calculating EV directly
Beyond just knowing whether a call is profitable, you can also calculate how much money you expect to win or lose.
Using the first scenario ($50 pot, $32 shove):
$$\text{Total pot if you call}=$114\\EV=(0.174*$114)-$32=-$12.16$$
So on average, you lose $12.16 every time you call here.
With the larger pot ($132 pot, $32 shove):
$$\text{Total pot if you call}=$196\\EV=(0.174*$196)-$32=+$2.10$$
This makes the call marginally profitable — you earn $2.10 per call in the long run.
Building Intuition
With practice, you won’t need to crunch the numbers every time. Repetition helps you quickly recognize when a situation is +EV (profitable) or –EV (unprofitable). In no-limit hold’em, there are countless pots and betting spots. Each time you watch someone bet, try working out their opponent’s pot odds and estimating how much equity they’d need to continue. Over time, this process becomes second nature.
In the follow-up to this article, we go a step further and explain the concept of Implied Odds
