If the odds are “fair”, then in a 2-horse race, if one horse is 10-1 on, the other would be 10-1 against. You can assume that this roughly reflects each horse’s chance of winning the race.

Bookies’ odds, of course, are not “fair” - if they were, they wouldn’t make any money. You can see this in the above example: by betting £10 on the horse at 10-1 on, and £1 on the horse at 10-1 against, you break even whichever horse wins (apologies if this is too simplistic).

This means that if, in a two-horse race, one horse is 10-1 on, the other horse would probably be (say) 7-1 against. This gives the bookie a profit margin, which they can increase (by making the outsider 5-1 against) or decrease (9-1 against).

Similarly, in a 3 horse race where the favourite is 10-1 on, and they other two horses are equal, they could both be priced at 4-1 against. If one is significantly better than the other, the 2nd best could be 2-1 against and the outsider 6-1 against.

Of course, if you are actually a bookie, other factors may come into play, such as wanting to attract more punters on the outsider. I think I have heard of cases where bookies actually offer far longer odds than what they assess the actual chance of winning to be, to attract some money away from the favourite and to lay off other bets. I may be wrong about this, however.

The maths involved in working out the above is made a lot easier if the decimal odds system is used (i.e. evens is 2, 5-2 against would be 3.5, etc.), but I’ll leave the explanation of that to someone else as my understanding of it is sketchy to say the least.