- The New England Patriots are playing the Buffalo Bills.
- The fractional odds are Patriots 1/2 and the Bills 2/1, where a bet on the Patriots risks $2 to win $1, and a bet on the Bills risks $1 to win $2.
- A rational bettor has $1 to bet.
- A $1 wager on the favorite Patriots yields a $0.50 profit, if they win.
- Or, a $1 wager on the underdog Bills yields a $2.00 profit, if they win.
- Therefore, the relative payout on the Bills is 4X that of the Patriots (2.00/0.50; this simple ratio mathematically ties the teams’ odds together, making the odds-to-probability relationship nonlinear).
- As odds-and-probs have an inverse relationship, the probability that the Bills win is 1/4th that of the Patriots.
- Therefore, the Bills have a 20% chance of winning, which is 1/4th that of the Patriots 80% chance of winning.
The conventional conversion of odds-to-probability is inverse-linear, where the probabilities are derived independently for each outcome from its odds. This overstates the probability of the underdog, while understating the probability of the favorite ... hence, the well-known 'longshot bias'.