Odds to probability - include field size and market dominance

Question

I am trying to understand the conversion of odds to probability. Reading on the internet there seems to be many references to calculations like this.

Horse Odds* Probability (from odds) ----- ---- ----------------------- Horse A 1:1 1/(1+1) = 1/2 = 6/12 Horse B 1:2 1/(1+2) = 1/3 = 4/12 Horse C 1:3 1/(1+3) = 1/4 = 3/12 Horse D 1:5 1/(1+5) = 1/6 = 2/12 ----- Total Probability = 15/12 > 1

from straight odds conversion

However to me this seems incorrect as the calculation doesn't:

1. Account for number of variables in the event I would contend that a horse in a 2 horse race with 1:1 odds has a different probability to a horse with 1:1 odds in a 15 horse race.
2. They do not account for market weight. Let me explain what I mean by this

Comparing 2 * 5 horse races

Race 1           Race 2
1. $2.50         1. $2.50
2. $3.25         2. $4.50
3. $7.00         3. $5.00
4. $11.00        4. $11.00
5. $21.00        5. $13.00

Race 1 spread of betting is wider compared to Race 2 so I am saying that the weight of the market is heavier for Race 1 number 1 compared to Race 2 number 1 as more money is directly "focused" on 1 runner causing others to have longer odds.

Therefore to be able to convert the odds into more reflective probabilities including size of the field and market weight or market dominance of an individual how can I do this? Is there a way to use standard deviation?

Answer 1

You appear to be mistaken about the meanings of "odds" and "probability" here. As used in this instance, neither one has anything to do with how much money has been bet or will be paid out.

Rather this calculation is only about how likely it is that a horse will win the race. By "odds", they mean:

If the same race could repeated a large number of times (without any changes in condition of the track, horses or their riders), then the odds for a horse is the ratio of the number of times the horse wins to the number of times it loses.

And by "probability", they mean:

If the same race could repeated a large number of times (without any changes in condition of the track, horses or their riders), then the probability for a horse is the ratio of the number of times the horse wins to the total number of races.

(In both of these definitions, I am ignoring the possibility of a tie.) This is the reason for the formula: Let

• $N$ be the total number of races.
• $W$ be the total number of times the horse wins.
• $L$ be the total number of times the horse loses. Then $N = W + L$.

The odds are given by:$$\text{Odds } = W\ :\ L = \frac WL$$ While the probability is given by $$\text{Probability } = \frac WN = \frac{W}{W+L}$$

Of course, the same race cannot be run multiple times, nor can anyone actually know what the outcomes would be if it were. So the odds or probability you are given are just someone's educated guesses.