Say we have three runners: A, B and C, and we have the probability of each runner beating each individual opponent:

A before B: 0.68

A before C: 0.42

B before A: 0.32

B before C: 0.30

C before A: 0.58

C before B: 0.70

Of course, the probability of A before B is = 1 - (probability B before A)

How would I go around calculating the odds of a given runner winning the race?

I thought I could consider the individual odds independent from each one, so I could just multiply them. Say:

Odds of A winning = P (A before B) * P (A before C) = 0.68 * 0.42 = 0.286

But then if I calculate the odds for B and C I get:

Odds of B winning = P (B before A) * P (B before C) = 0.32 * 0.30 = 0.096

Odds of C winning = P (C before A) * P (C before B) = 0.58 * 0.70 = 0.406

The three odds should come to 1, but it doesn't add up:

0.286 + 0.096 + 0.406 = 0.788 (not = 1)

What is it I am doing wrong?

Thanks!