In horse-racing, as already outlined in Small Fields And Exotic Convex Bets (Part 1), we can look for better prices in the exotic markets when we are faced with small fields of runners.
On this occasion, we will focus on trifecta bets in eight and nine-runner fields.
As before, using an Excel spreadsheet, we can set up a table as follows to generate trifecta tickets subject to the following conditions:
- Runner saddle numbers as both row and column labels,
- Populate the table
with 1s and 0s subject to the following constraints:
- Row totals must equal three.
- Column totals must equal three.
- Use Solver to generate a possible solution. Otherwise, use trial and error.
- Read off the column labels of the rows containing 1s to generate a minimum set of tickets.
- There are multiple valid combinations of tickets.
- No matter what is the winning combination of the race, the generated set will contain one ticket with at least two of the winning numbers (not necessarily in correct order).
This is an example of generating a trifecta solution for an eight-runner field.
and this is an example of generating a trifecta solution for a nine-runner field.
Ideally, to generate a valid set of tickets that meet our constraints, we should use solver (3-5 mins).
The following tables show the probabilities of getting one or more tickets with at least two of the winning numbers (not necessarily in correct order) For example, we have a 23% chance of getting one such ticket in a nine-runner field.
| CombCount | Count | Probability |
|---|---|---|
| 0 | 0 | 0% |
| 1 | 19 | 23% |
| 2 | 44 | 52% |
| 3 | 20 | 24% |
| 4 | 1 | 1% |
One possible approach is, having selected a positive expected value trifecta combination, that we generate a random, solver set to cover the remainder of the field.
Enjoy!