Calculators every bettor should bookmark

Most bettors carry five recurring arithmetic problems and solve none of them cleanly. What does this parlay pay, and is that price fair? How much goes on the other side to come out even? Is this middle worth chasing? What is a $100 free bet actually worth? What is the true price behind the number on screen? Sharp ships a calculator for each. Here is the formula behind all five, a worked example you can check line by line, and the judgment call each one doesn't make for you.

The two conversions underneath all five

A calculator is a speed device, not a thinking device. Know the formula and it saves you forty seconds; don't, and it hands you a number you can't sanity-check.

American to decimal: negative odds → 1 + 100/(-O). Positive → 1 + O/100. So -110 is 1.9091, +150 is 2.50.
American to implied probability: negative odds → -O/(-O + 100). Positive → 100/(O + 100). So -110 implies 52.38%, +150 implies 40.00%.

Everything below is those two plus multiplication.

1. Parlay — multiply the decimals

The question: what does this ticket pay, and is the offered price fair?

The formula: convert every leg to decimal, multiply, and the product is the parlay's decimal price. Back to American with (d − 1) × 100 when d ≥ 2.

Worked example. Three legs: -110, +150, -200.

1.9091 × 2.5000 × 1.5000 = 7.1591 decimal, or +616
A $100 ticket returns $715.91. That price implies 13.97%

Now the part the payout box hides. Devig each leg against its opposite side:

-110 / -110 → 52.38% + 52.38% = 104.76% → fair 50.00%
+150 / -180 → 40.00% + 64.29% = 104.29% → fair 38.36%
-200 / +165 → 66.67% + 37.74% = 104.40% → fair 63.86%

Multiply the fair probabilities: 0.50 × 0.3836 × 0.6386 = 12.25%. Fair decimal is 8.166, or +717. You're offered +616 on something worth +717. EV per $100 is 0.1225 × 7.1591 − 1 = −12.33%. That's the parlay hold — three roughly 4.5% markets compounding into one expensive ticket.

Where naive multiplication breaks. Multiplying assumes independence, and same-game legs almost never are. Say a quarterback's passing-yards over is a fair 50% and his top receiver's yards over is a fair 50%. Naive multiplication says 25%, or +300 fair. But if the quarterback goes over, the receiver's over gets far likelier — call it 62%. True joint is 0.50 × 0.62 = 31%, or +223 fair. At an offered +240 (decimal 3.40), the naive read says you're being robbed; the correlated read says 0.31 × 3.40 − 1 = +5.4% EV.

Reverse it and you get the trap. Pair a quarterback over with his own team's under: if the conditional drops to 36%, true joint is 18% and +300 is −28% EV on a ticket that looks fairly priced. See [#17 — Correlated parlays — when SGPs are actually +EV].

2. Hedge — the price of certainty

The question: how much on the other side to return the same amount either way?

The formula: H = (S × d_original) ÷ d_hedge.

Worked example. You hold $200 at +600 (decimal 7.00) on a title future. Your team reaches the final; the opposing side is +120 (decimal 2.20).

Your ticket returns $200 × 7.00 = $1,400
Hedge stake: 1,400 ÷ 2.20 = $636.36
Original wins: $1,400 − $200 − $636.36 = +$563.64
Hedge wins: $636.36 × 2.20 = $1,400, − $200 − $636.36 = +$563.64

What it costs. Devig the final: -140 / +120 implies 58.33% + 45.45% = 103.79%, so your side is a fair 56.20%. Holding is worth 0.5620 × $1,400 = $786.86 expected. Hedging fixes $763.64. The hedge costs $23.22, or 2.95% of expected value — essentially the vig on the hedge leg. That's the whole trade: pay the hedge market's vig to convert a variable outcome into a fixed one. [#48 — Hedging math — when it makes sense, when it costs you EV] is where that call gets made.

3. Middle — the gap between two numbers

The question: the line moved after I bet it. What does taking the other side cost, and how often must the middle hit?

Worked example. You took an underdog at +7 (-110). The line moved; you take the favorite at -3 (-110). Both tickets $110 to win $100.

Favorite by 4, 5 or 6: both win, +$200
Favorite by exactly 3 or exactly 7: one pushes, one wins, +$100
Anything else: −$10

The cost of the middle is that −$10 — the vig on the losing side, not the stake. Which is why middles are cheap to attempt and easy to over-attempt.

Breakeven: p × 200 = (1 − p) × 10, so p = 10 ÷ 210 = 4.76%. If you believe it lands 8% of the time, EV is (0.08 × $200) + (0.02 × $100) + (0.90 × −$10) = +$9.00 on $220 at risk — a 4.09% ROI.

Price sensitivity is brutal. Take both sides at -115 instead: the split costs $15, so breakeven jumps to 15 ÷ 215 = 6.98%, 45% more middle frequency for five cents of extra juice. [#46 — Middling — when to fish for the gap and what it actually costs] covers which gaps earn the attempt.

4. Free-bet conversion — face value is not cash value

The question: what is a promotional credit worth in dollars?

A free bet returns profit only; the stake doesn't come back. A $100 free bet at decimal d pays $100 × (d − 1) on a win.

The ceiling formula: in a vig-free market, expected cash value as a fraction of face is 1 − 1/d. That one expression explains the whole longshot rule.

Placed at	Decimal	No-vig ceiling
-200	1.500	33.3%
+100	2.000	50.0%
+200	3.000	66.7%
+400	5.000	80.0%
+600	7.000	85.7%

The longer the price, the less of the credit's value sits in the stake you never get back. That is the entire mechanism.

Worked example, hedged. $100 free bet at +400 (5.00), hedged against -450 (1.2222):

Free-bet profit if it wins: $400
Hedge stake: 400 ÷ 1.2222 = $327.27
Longshot wins: $400 − $327.27 = +$72.73
Favorite wins: $327.27 × 0.2222 = +$72.73

72.7% conversion. Run it across the practical range and the rule of thumb falls out: +150 hedged at -165 → 56.6%; +200 at -220 → 62.5%; +300 at -340 → 68.2%; +400 at -450 → 72.7%. Most two-way markets with real liquidity sit between +150 and +300, which is why 60-70% of face value is the standard free-bet conversion benchmark. Full workflow in [#14 — Promo conversion — turning sportsbook offers into cash].

5. True odds — devig, then convert back

The question: stripped of the book's margin, what is fair?

The formula (equal-multiplier): fair p(A) = implied A ÷ (implied A + implied B). Then decimal = 1 ÷ p, and American = (d − 1) × 100 if d ≥ 2, else −100/(d − 1).

Worked example. A reference book posts -145 / +125.

-145 → 59.18%; +125 → 44.44%; sum 103.63%, so 3.63% hold
Favorite: 59.18 ÷ 103.63 = 57.11% → decimal 1.751 → -133
Underdog: 44.44 ÷ 103.63 = 42.89% → decimal 2.332 → +133

Now shop. Another book posts the underdog at +145 (decimal 2.45, implied 40.82%). Your fair estimate says 42.89%. EV = 0.4289 × 2.45 − 1 = +5.08%, or $5.08 per $100.

Equal-multiplier is the Sharp default, with a known weakness on heavy favorites and longshots where it slightly under-prices the favorite. [#4 — How Pinnacle's no-vig odds actually work] is the foundation; [#7 — Devigging methods compared — which to trust when] is when to switch.

The order to run them in

A sequence, not a menu. True odds first — everything else depends on a fair-price estimate, and without one you're comparing offered prices to other offered prices. Parlay math second if the ticket has legs, correlation check included. Free-bet conversion whenever promotional credit is in play, because face value is a marketing number. Middle and hedge last, since both are post-bet decisions on a position you already hold, and both spend EV to buy something else — certainty, or a shot at winning twice.

None of them tells you what to do with the answer. They tell you what the number is. Bankroll sizing still applies, the 5% single-bet cap still applies, and the decision is still yours.