Box 3 or Key?

by Steven G. Berry

It's the age-old question, isn't it? Should I box my top choices or key the probable winner over a multitude of contenders? Experts have argued both approaches for many years

If we are going to invest the same amount of money boxing or keying, which is the better strategy?

Let's look at trifectas for sake of argument. Trifectas are midway between exactas and superfectas. You can easily contract or expand the facts to fit the other exotics. This article assumes using the minimum horses in a box versus the same amount of money keying your top selection to win.

I believe that the key to understanding your chances of winning exotics is based upon your ability to select winners. For sake of argument, let's say you pick 30 percent winners. We may agree that your odds of selecting a winner vary depending on the tote odds of your horse and how many other horses are entered in the race. In general, lower odds increase your chance of winning while more entries decrease your chance. For this discussion, we will omit odds and field size and concentrate on an overall win percentage of 30%.

If you pick the winner in an exotic, what is your chance of correctly selecting the second horse? Your second choice is now your selection to “win” second so we will assume he has a 30 percent chance. The same theory holds while selecting the third finisher, who has a 30% chance of beating the remainder of the field for the show.

From the above, we know that if we bet a straight $1 trifecta we have a 30% chance of running first, a 30% chance of running second and a 30% chance of running third. Since we need to nail win, place and show together, we must multiply our odds to determine our chance of winning.

30% times 30% times 30% = 2.7%

That's not much hope is it? So, we throw more money into the play in order to increase our chances. Adding a second horse doesn't double your chances, however.

Consider adding a second horse to win. Does that give you a 5.4% chance? No. Your top selection should win 70% of the time, so your second selection, now picked to run first, has only 70% of race instances in which to run first 30% of the time.

If your top choice has a 30% chance of winning, your second choice has a 30% chance of winning whenever your top choice loses. Your top choice loses 70%, so the second choice must win 30% of the 70% of the races in which your top choice does not win. Your second choice wins 21% of the time.

Admittedly, this is a little confusing, but the idea can be extended to deeper choices who each win 30% of the races that the horses superior to them did not win. This idea results in an extremely useful table that can be used in figuring odds on a great number of exotic bets:

Approximate chance of selections winning race with 30% top winners:

1st choice: 30%
2nd choice: 20%
3rd choice: 15%
4th choice: 10%
5th choice: 8%
6th choice: 5%
7th choice: 4%
8th choice: 3%

This table will prove useful for calculating exotic odds. If you have only six dollars to invest, what are your trifecta choices?

1 over 2 over 3,4,5,6,7,8

In this scenario, you have a strong opinion on the exacta. Your top two horses look very strong so you key them both over many horses.

At first glance, it seems we have many horses running for us. Unfortunately, all those horses for third only matter if you hit the exacta and your chance of that is 30% time 30% or 9%. Ninety-one percent of the time, your bet is dead.

If you do get the first two, you have six horses running for third. When calculating for third, consider the first two as “gone” and use the odds for first through sixth. You have an 88% chance of running third with one of your six remaining horses.

So, with a 9% chance of the exacta followed by an 88% chance for third you are left with about an 8% chance of winning.

As we shall see, that's a small chance for a $6 investment. Also, it is human nature to play low odds horses as the top two selections which will usually result in a low trifecta payoff.

1 over 2,3 over 2,3,4,5

A lot of people like this trifecta play. It keys your best horse over your next two best horses and then gives you three horses to run third should you nail the first two levels.

It appears to give you a lot of horses running for your money, you play through the fifth choice!

Here's your chance to win: 30% times 50% times 65% = about 10%.

First 30% = 30% *
Second 30% + 20% = 50% *
Third 30% + 20% + 15% = 65%
= 9.75% let's call it 10%

If you hit one of ten trifectas, you need to average a $60 payoff to break even. You will bet $600 in 100 races and win 10 times. Sixty dollar trifectas seem trivial until you start collecting $16 trifectas.

It is human nature for players to use this betting structure when the top selection is low odds. The tote board reassures us that our “winner” will actually win. If follows that the average payoff will suffer.

1 over 2,3,4 over 2,3,4

This play is similar to the above and running the numbers you will find the odds are the same:

First 30% = 30% *
Second 30% + 20% + 15% = 65% *
Third 30% + 20% = 50%
= 9.75% let's call it 10%
Instead of playing an additional horse to run third, we have moved a horse into second. I think overall this will pay more money as the win and place positions affect exotic payoffs more than the lower positions. Consider an exacta with a longshot over the favorite instead of vice versa. The higher you can place a longshot the better your payoff.

There are other combinations you can make for $6, but a little study will show they have similar or worse odds to what has been covered. For example 1 over 4,5 over 2,3,4,5 seems a silly bet as you are now overlooking two decent horses for second meaning 30% * 25% * 75% or 5.6% total.

Box 3

Beginners love the box. Maybe this has something to do with “beginner's luck.” Boxing three horses in the trifecta costs $6, same as the examples above. Beginners get intimidated with all the possible combinations and so box out of frustration. I think this is to their advantage.

Experts will argue, “Why put as much money to win on your third choice as you do your first?”

What they forget is they are spending money on horses they don't expect to run in the money!

Why put trifecta money on a horse you don't expect to be in the trifecta? Seems like a waste to me.

I'd rather bet the three horses I expect to be on the wire than throw money at horses I expect to run up the track somewhere. The deeper you reach, the more you rely on luck. I'd rather be lucky than good, but you can't handicap luck. Are you handicapping or gambling?

The box is just as strong mathematically as any top key:

First 30% + 20% + 15% = 65% *
Second 30% + 20% = 50% *
Third 30% = 30%
= 9.75% let's call it 10%
The beauty of the box is you are only betting the three horses you actually expect run well and they can run in any order.

Sometimes a top three pick has long odds and sometimes they win.

Because of the human nature tendency to key a favorite rather than a longshot, boxing should provide a better profit than keying a top horse for most people.

Adding another horse

What if we box four horses in the trifecta instead of three? That next horse is going to cost you dearly.

We figure trifecta costs by the following formula:

NumHorses * (NumHorses – 1) * (NumHorses – 2)

With a 4 horse box that is 4 * 3 * 2 = $24 or four times the cost of a 3 horse box!

Do your chances quadruple? Nope:

First 30% + 20% + 15% + 10% = 75% *
Second 30% + 20% + 15% = 65% *
Third 30% + 20% = 50%
= 24.38% let's call it 25%
You go from one in ten hits to one in four hits or two and one-half times a better hit rate, not four.

It is obvious that you'll need higher payoffs to break even.

At $6 bets, you need $60 trifectas to break even. ($600 / 10 in 100 plays.)

At $24 bets, you need $96 trifectas (average) to break even. ($2400 / 25 in 100 plays.)

Adding yet another horse

What if we box five horses in the trifecta?

Now the cost is 5 * 4 * 3 = $60. Yikes! Ten times our 3 horse box for 4 times the chance.

Five horse Trifecta box:

First 30% + 20% + 15% + 10% + 8% = 83% *
Second 30% + 20% + 15% + 10% = 75% *
Third 30% + 20% + 15% = 65%
= 40.46% let's call it 40%

At $6 bets, you need $60 trifectas to break even. ($600 / 10 in 100 plays.)

At $60 bets, you need $150 trifectas (average) to break even. ($6000 / 40 in 100 plays.)

Conclusion

Boxing the minimum versus keying at equal investment favors boxing.

Boxing beyond the minimum box plus one horse relies too much on luck and most likely a superior chance can be constructed from keying one or even two horses. I guess it all depends on how many combinations you are willing to play.

It should be mentioned, that the most cost-effective bet is to play ONLY your top selection:

1 over 2 over 3:

(30% * 30% * 30% = 2.7% for $1 or 1 unit.) $6 is not worth 2.7% * 6 = 16.2%