Implied Probabilities and Bookmaker Overround
A sportsbook quotes decimal odds of $O_A = 1.72$ on Team A and $O_B = 2.35$ on Team B for a game where exactly one team wins.
- Compute the implied probability of each outcome from the posted odds.
- Calculate the bookmaker's overround (also called the "vig" or "juice") -- the margin baked into these quotes.
- Determine the no-arbitrage range of "fair" win probabilities for Team A that are consistent with these two quotes.
Hints
- Decimal odds $O$ represent the total payout per unit staked. The implied probability is simply $1/O$ -- the break-even win rate for the bettor at those odds.
- Sum the two implied probabilities. If the total exceeds 1, the excess is the bookmaker's overround (margin). This is the vig that keeps books profitable on balanced action.
- For the no-arbitrage bounds, set the expected profit of betting each side to zero and solve for $p_A$: betting A gives $\pi_A = p_A \cdot O_A - 1 \leq 0$, betting B gives $\pi_B = (1-p_A) \cdot O_B - 1 \leq 0$. Each constraint gives one endpoint of the range.
Worked Solution
How to Think About It: Decimal odds tell you the payout per unit staked, including stake return. So odds of $O_A = 1.72$ mean: bet \$1, get back \$1.72 if Team A wins. The implied probability is just $1/O_A$ -- the break-even probability if the book had zero margin. But if you sum up $1/O_A + 1/O_B$ and get a number greater than 1, the book is extracting that excess as profit. That excess is the overround. The no-arbitrage range is then the interval of true Team A probabilities where neither betting on A nor betting on B at these odds has positive expected value for the bettor.
Quick Estimate: Let's ballpark. $1/1.72 \approx 0.58$ and $1/2.35 \approx 0.43$. Sum is about $0.58 + 0.43 = 1.01$ -- so the overround is roughly 1%, or about 1 cent on the dollar. That is a lean book. Typical sportsbooks run 3-10%, so this is a tighter-than-average market. Quick sanity: the fair probability for A must be somewhere below 0.58 (otherwise betting A is a freeroll for bettors). Similarly it must be above $1 - 0.43 = 0.57$. So the fair probability for A is pinned in a very narrow band around 57-58%.
Approach: Work directly from the definition of decimal odds. No arbitrage requires that the expected profit from betting either side at the quoted odds is non-positive.
Formal Solution:
Step 1 -- Implied Probabilities.
Decimal odds $O_i$ imply a break-even probability (from the book's perspective):
$$p_A^{\text{impl}} = \frac{1}{O_A} = \frac{1}{1.72} \approx 0.5814$$
$$p_B^{\text{impl}} = \frac{1}{O_B} = \frac{1}{2.35} \approx 0.4255$$
Step 2 -- Overround.
The overround $\Omega$ is the sum of implied probabilities minus 1:
$$\Omega = \frac{1}{O_A} + \frac{1}{O_B} - 1 = 0.5814 + 0.4255 - 1 = 0.0069$$
So the book is running roughly a 0.69% margin on this game. Equivalently, for every \$100 of balanced two-sided action, the book expects to keep about \$0.69.
Alternatively expressed as a percentage of total implied probability: the overround is $\Omega / (1 + \Omega) \approx 0.68\%$ of handle. Both conventions appear in practice -- know which one your interviewer means.
Step 3 -- No-Arbitrage Range for $p_A$.
Let $p$ be the true probability that Team A wins. The bettor's expected profit from a \$1 bet on A is:
$$\pi_A(p) = p \cdot (O_A - 1) - (1 - p) \cdot 1 = p \cdot O_A - 1$$
No-arbitrage on the A side requires $\pi_A(p) \leq 0$, i.e.:
$$p \leq \frac{1}{O_A} = 0.5814$$
Similarly, a \$1 bet on B has expected profit $\pi_B(p) = (1-p) \cdot O_B - 1$. No-arbitrage on the B side requires:
$$(1 - p) \leq \frac{1}{O_B} \implies p \geq 1 - \frac{1}{O_B} = 1 - 0.4255 = 0.5745$$
So the no-arbitrage range for the true Team A win probability is:
$$0.5745 \leq p_A \leq 0.5814$$
This interval has width exactly equal to the overround $\Omega \approx 0.0069$. The book can only be arbitrage-free if the true probability falls inside this band.
Answer: - Implied probability of A: $p_A^{\text{impl}} \approx 58.1\%$ - Implied probability of B: $p_B^{\text{impl}} \approx 42.6\%$ - Overround: $\Omega \approx 0.69\%$ - No-arbitrage range for Team A: $p_A \in [0.5745,\ 0.5814]$, a band of width equal to $\Omega$
Intuition
Decimal odds are just compressed probability estimates with a margin layered on top. Every sportsbook, prediction market, and options desk does the same thing: quote prices that sum to more than 1 (or, equivalently, quote bid-ask spreads) so that balanced two-sided flow generates a risk-free profit. The overround is the sports analogue of the bid-ask spread on an options market -- it is the price you pay for immediacy and liquidity. A 0.69% overround is unusually tight; recreational sportsbooks often run 4-8%, and regulated bookmakers in certain markets run 10% or more. Knowing how to back out the overround instantly is a basic sanity check when looking at any quoted odds.
The no-arbitrage range is the key practical insight. It formalizes the idea that the book does not need to know the true probability exactly -- it just needs to post odds wide enough that no bettor can guarantee positive expected value regardless of the true outcome. The width of that safe zone is exactly the overround. This same logic underpins option market making: a market maker posts a bid-ask spread wide enough that, regardless of which direction the stock moves, the flow does not systematically exploit them. The bigger the overround (or spread), the more protected the book (or market maker), but the less attractive the market is to bettors (or traders).