Expected value (EV)

Expected value (EV) answers a simple question: if I placed this bet thousands of times, how much would I win on average per euro staked?

The formula

For a stake of 1 unit:

where $p$ is your estimated probability.

• EV > 0: profitable bet in the long run (+EV)
• EV = 0: neutral bet
• EV < 0: losing bet in the long run (−EV)

EV and ROI

EV is a forecast. ROI (return on investment) is the actual result:

Over a very large number of bets, if your estimates are accurate, your ROI converges towards your average EV. Over a small number of bets, the two can be very far apart (that's the topic of the intermediate level).

Key takeaways

• EV depends entirely on the quality of your $p$. An EV of +12.5% based on a bad estimate is worthless.
• A small EV repeated thousands of times beats a big EV played three times.
• An EV that's "too good to be true" (+30%, +50%) is often a sign of an error: stale odds, bad information (injury, line-up), or a wrong model.