Odds of winning a race given the odds of each runner beating each opponent?

#1
Hi,

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!
 

BGM

TS Contributor
#2
I thought I could consider the individual odds independent from each one, so I could just multiply them.
No they are not independent.

Actually it is very natural. E.g. given A beating B, we know that A has a better performance and actually this improve the odds that A beating C also.
 
#3
BGM, thanks for your replay.

Ok, I sort of suspected it.

So, is there a way to calculate the odds for a runner winning, given the information that we have?