No reverse gear
Harville's formula — pricing combinatorial bets by repeated renormalisation of winning probabilities — is the classical application of Luce's Axiom of Choice to racing. It is also logically inconsistent: the same event gets different probabilities depending on whether you compute it forward (from the winner) or backward (from the loser). Try it.
Winning probabilities
Set the three winning probabilities. The defaults reproduce the canonical example $(1/2, 1/3, 1/6)$ from the book.
$p_3 = 1 - p_1 - p_2$. We auto-rescale if your sliders push $p_3$ negative.
Forward Harville
$P(\text{H1 first, H2 second})$, computed winner→runner-up:
$P(\text{H1 first}) \times \dfrac{p_2}{1 - p_1}$
Backward Harville
Same event, computed loser→runner-up-from-last using losing probabilities (themselves obtained from Harville forward):
$P(\text{H3 last}) \times \dfrac{p_2^{\text{lose}}}{p_1^{\text{lose}} + p_2^{\text{lose}}}$
The damning ratio
These two numbers describe the same event: horses 1 and 2 take the top two positions in that order. If a model is coherent the ratio is 1. Harville gives:
You can chase the discrepancy away by setting $p_2 = p_3$ — then the field becomes symmetric in those two, and forward and backward Harville agree by accident. The only generative setup that makes them agree everywhere is the one where performance distributions are exponential (i.e.\ Luce's axiom holds by force of the distribution). For any other model, repeated renormalisation is incoherent — running the same arithmetic backward from the loser instead of forward from the winner gives you a different answer.
The Thurstone alternative computes the probability of any finishing arrangement from the underlying performance density once, with no asymmetry between “the winner is drawn first” and “the loser is drawn last”. See the fast ability transform page for how, or the forward pricing demo for it in action.