expected-value-basic draft

Expected value (EV) for a single bet is the average dollars-per-dollar you'd win or lose if you placed the same bet many times at the same price, against your assumed probability of winning. It's a long-run quantity — a per-trial mean — not a guarantee about any one bet.

For a unit stake at fair probability p and decimal odds d, the standard formula is:

EV_per_dollar = p × (d − 1) − (1 − p)
            = p × d − 1

The two forms are equivalent. The first says "with probability p you win (d − 1) dollars (the profit), and with probability (1 − p) you lose 1 dollar (the stake)." The second is the same thing rearranged: p × d is your average gross return per dollar staked, and subtracting the stake itself leaves the average net.

Kairos's kairos/core/ev_edge.py::ev_pct_vs_vf_target implements exactly this: it converts the "VF target" (devigged American odds for the side you're considering) to a fair probability, converts the price you're paying to decimal odds, and computes (p_fair × decimal_odds − 1) × 100 as a percentage. This is per-stake ROI — return on the dollar wagered.

Sign convention

• Positive EV means a favorable bet: you expect to win money on average.
• Negative EV means an unfavorable bet: you expect to lose money on average.
• Zero EV is breakeven — you'd neither win nor lose over a long run.

A bet at the "fair price" (decimal odds equal to 1/p) has EV exactly zero. Anything tighter than fair is negative; anything looser is positive. Real sportsbook prices are negative-EV by construction once vig is included — that's how the book makes money — so positive EV against the raw book line is rare. EV against the devigged fair line, however, is the meaningful question for a sharp player. See vig-adjusted-ev.

Units

Mimir's canonical form is EV-per-dollar-staked, expressed as a fraction. So 0.05 means a 5% edge, or 5 cents of expected profit per dollar wagered. Equivalent expressions:

• Fraction: 0.05
• Percent: 5.0%
• Per-$100: $5.00 per $100 staked

Some downstream tools (and Kairos itself) sometimes display EV in absolute dollars on the actual stake size (EV $12.50 for a $250 bet at 5%). That's just the per-dollar EV multiplied by the stake — a presentation choice, not a different quantity. Per-dollar is the right basis for comparing bets of different sizes; absolute-dollar EV is the right basis for budgeting.

Scaling with bet size

EV-per-dollar is a constant for a given (price, probability) pair. EV in absolute dollars scales linearly with stake size:

EV_dollars = stake × EV_per_dollar

Doubling your stake doubles your expected dollar return — and doubles your variance. Choosing how much to risk given an edge is a separate question (see Kelly criterion, bet sizing — both forthcoming Mimir entries). The edge itself doesn't change with stake; only the dollar magnitude does.

Worked example

A binary moneyline shows Side A at +150. You believe (from a devigged consensus, or your own model) that A's fair probability is 0.45 — i.e. 45%. What's the EV?

• Decimal odds at +150: 1 + 150/100 = 2.50. (See american-decimal-conversion.)
• Fair probability: p = 0.45. Loss probability: 1 − p = 0.55.
• Apply the formula (first form):
• EV = p × (d − 1) − (1 − p)
• EV = 0.45 × (2.50 − 1) − 0.55
• EV = 0.45 × 1.50 − 0.55
• EV = 0.675 − 0.55
• EV = 0.125
• Equivalently (second form): EV = p × d − 1 = 0.45 × 2.50 − 1 = 1.125 − 1 = 0.125.

So the bet has EV = +0.125 = 12.5% per dollar staked. On a $100 stake, expected profit is $12.50 over a long run.

A useful sanity check: the price's implied probability is 1/2.50 = 0.40 (40%). Your fair probability is 0.45, which is 5 percentage points higher than the price implies — you think the bet wins more often than the price says it should, so EV is positive. The size of the EV depends on both the gap and the price level (the same 5-pp gap is worth more EV at long odds than at short odds, because the payout per win is larger).

Gotchas

• EV depends entirely on the p you supply. Garbage in, garbage out. A wrong fair probability produces a confident, precise, and meaningless EV number. The hard work in EV-based betting is producing trustworthy ps, not the EV arithmetic itself. Treat displayed EV as a function of an estimate, never as a measurement.
• EV against the raw vigged line is not the same as EV against the devigged fair line. They answer different questions; the first is almost always negative for a retail bettor; the second is what sharp pricing actually targets. See vig-adjusted-ev for the distinction.
• EV is a long-run expectation; single bets fluctuate widely. A 5% edge on a typical moneyline price has a per-bet standard deviation of nearly $1 per dollar staked. You can lose ten in a row and still be making correct +EV bets. Variance dominates short samples; only long runs (hundreds to thousands of bets) reveal the EV signal through the noise.
• EV is not the same as win rate. A bet at +150 with p = 0.45 is +EV but loses 55% of the time. Conversely, a heavy favorite at −500 with p = 0.80 is −EV (0.80 × 1.20 − 1 = −0.04) despite winning 80% of the time. Win rate and EV diverge whenever the price doesn't match the probability — which is the whole point of looking at EV.
• Don't confuse per-dollar EV with absolute-dollar EV when comparing bets. A $1,000 bet at 1% edge has $10 expected profit; a $50 bet at 8% edge has $4. The second is a better bet (more efficient use of capital and risk budget) even though the dollar EV is smaller. Per-dollar EV is the comparable quantity.
• EV at exactly zero is breakeven, not "no risk." A zero-EV bet still has full variance — you just don't expect to win or lose on average. Do not interpret zero EV as "safe."

Open questions

• Kairos's compute_ev_pct (in kairos/core/ev_edge.py) supports three denominator modes for converting absolute-dollar EV into a percentage: ROC (return on cost / total wager), ROR (return on risk), and ROB (return on base wager — min(stake, win) per side, summed across legs). These are presentation conventions for multi-leg or asymmetric bets, not different EV concepts. Mimir documents the per-dollar fraction-of-stake as canonical; whether Mimir should also surface the ROC/ROR/ROB distinction as a documented variant is open.
• The formula above assumes a single binary win/lose outcome. Pushes (a wager that voids and refunds the stake — common on point spreads landing on the number, totals at the exact total) shift the effective probability mass and require a three-way EV calculation: p_win × (d − 1) + p_push × 0 − p_loss × 1. Whether Mimir treats push-aware EV as a separate entry or a gotcha here is open.

Cross-references

• mimir: american-decimal-conversion
• mimir: american-implied-probability
• mimir: devigging
• mimir: vig-adjusted-ev
• mimir: vf-and-mba-targets