Originally Posted by
Justin7
Forget that. It wasn't quadratic.
Now, if you want something harder...
Best of 5 match - favorite is -150.
What is fair price for best of 3?
This is obviously fundamentally no different than the problem as posed above.
Let Px = probability of player winning a x-set match (for all odd integers x > 0)
Let P = P1 = probability of player A winning a single set match
Then in general:
[nbtable][tr][td]Px = P (x+1) 2 * [/td][td][/td][td]{ (1-P) (2i-x-1) 2 * combin(i-1, (x-1) 2 ) }[/td][/tr][/nbtable]
So given x = 5 and Px = 150 250 = 60%:
P5 = P * ( 1 + (1 - P) * 3 + (1 - P)2 * 6) = 60%
Solving the quintic equation for P yields a single real root of P ≈ 55.375%.
Plugging back into the summation above:P3 = P2 * ( 1 + (1 - P) * 2)
P3 ≈ 55.375%2 * ( 1 + (1 - 55.375%) * 2)
P3 ≈ 58.031%
For a fair money line of roughly -138.3.
Note that we can also obtain the same probability via the Binomial distribution:P3 = 1 - BINOMDIST(2-1, 3, 55.375%, 1)
P3 ≈ 58.031%
The very simple attached spreadsheet will determine answers to problems such as these using the Binomial Distribution+Solver.