What is Probable? (Part 1)

How well do we understand Probability? What is Probable? Most of us think we have a fairly good idea of what sort of chance a selection we have made might perform in a race, especially if we have taken the time to incorporate ideas and principles from ‘expert’ commentators. We hold to these convictions quite firmly, until our results cause us to wonder why we’re not getting ahead as our understanding of Probability would lead us to believe we should.

In fact, there are many examples of common failures to understand and assess Probability and some of them lead us to make amazingly inaccurate calculations about what seem, on the face of it, to be relatively simple proposals. Let’s consider a few of these.

First, let’s take the Common Birthday Conundrum. If we place thirty people in a room, what is the probability that any two of them will share the same day and month as birthdays? Most people will quickly reason along the lines of:

“Hmm, 365 days in the year, 30 people; must be about 1-in-10 or 12”.

When the Probabilities are correctly calculated (I won’t go into it here), the actual chance is over 70%! Most people simply won’t believe it, but it’s a mathematical fact. You could make a living from it if you went to enough parties, and took bets!

Second: The Pick-a-Box Paradox. In a TV game, contestants had to pick one of three boxes, for a chance of a prize. The Host would show them a losing box, and then ask if they wanted to change their pick. Overwhelmingly, people would stick with their original choice, and overwhelmingly, they would lose. Why? No doubt psychology played a big part; we all like to think our judgement of a 50-50 chance is as good as anyone else’s. But is it a 50-50 chance?

The contestants overlooked a simple fact. Whenever we pick from a 1-in-3, the odds of making the right choice is 2-1 against. If the contestants picked the winning box first, clearly they would lose by switching. However, the chance of winning straight off is only one in three. If the Host shows them a losing box, the Probabilities swing around. Once the Contestant knows for certain one of the losing boxes, their Probability of winning by switching is now two in three. They would have won twice as often by simply considering the Probabilities and changing their minds.

This runs counter-intuitive to most peoples’ way of thinking. Research shows that the vast majority of people think that once one box has been eliminated, the chance of picking the right one is still 50-50. I’d bet that most folk reading this would think the same (about 90% do).

So that I don’t get a lot of emails, I’ll explain it a little more fully.

If the first pick is wrong (2/3 chance that it is), switching must have a 2/3 chance of winning, since the other loser can no longer be picked. Another way of considering it is to think of the two unchosen boxes as one. These two must have a 2/3 chance of containing the prize. Since one of them is shown to be a loser, the other must have the 2/3 chance of being the winner, so a rational person would switch their choice from a 1/3 chance to 2/3, would they not? If you still don’t believe, try it with three dice or cards (looking for an Ace or a six): you will win twice as often by switching than by sticking.

Cognitive psychologist Massimo Piatelli-Palmarini stated: “No other statistical puzzle comes so close to fooling all the people all the time. Even Nobel physicists systematically give the wrong answer, and insist on it, and they are ready to berate in print those who propose the right answer”. * Furthermore, experiments with pigeons using this problem show that they rapidly learn to always switch, unlike humans! What chance do we have, when even the bird-brained can outperform us on a simple probability test!

So how are we going to do assessing the Probabilities in a horse race, with myriads of possibilities, not just three, to consider? To return to a favourite (pardon the pun) theme of mine, we know that market favourites only win about 1/3 of the time. Therefore 2/3 or more of winners come from those not considered favourite. There’s our first clue; since favourites are also underpriced (by virtue of being overbet), we should only occasionally back a favourite. If we fall into this habit, of consistently backing favourites, we will almost certainly lose over time. Whenever I hear someone say “The favourite looks good” or some such comment, I take it as the leper’s bell of an approaching punting calamity.

Even the real champions occasionally get beaten, and even if that is a rarity, the reward we get for backing them several times successfully will not adequately compensate for the time when we take the short price and lose heavily. Failure to grasp this concept is very akin to the examples given above; it’s a human frailty in assessing Probability; and in my opinion is responsible for more punters going bankrupt than any other factor. One only has to observe the wild cheering that occurs whenever a favourite wins the last race on the card, to strongly suspect that most punters have backed it out of desperation, hoping to ‘get out’ for the day.

It is surely better to conserve our capital by wagering smaller amounts on bets that will pay well, than to bet heavily at short odds, risking serious damage to our banks if we miss just a few attempts. In seeking to get value, we need to look for factors that others are ignoring or undervaluing in their assessments.

We also need to factor in more than most people, the element of luck. This is the biggest factor in any single race. The desire to eliminate the negative aspects of luck is what drives most punters towards favouritism. All the normal variables considered, (form, class, weight, barrier, track, fitness etc) are all efforts to minimize the influence of luck and increase the predictability of the outcome. Observation of even a few dozen races however, shows that the thousands of things that happen during a race render it virtually unpredictable. How else to explain the massive variation in not only the dividends paid for winners, but the apparent gulf in the differing abilities of winners as well? Of course 20-1 shots don’t get up anywhere near as often as top-class horses, but unless we’re going to bet on all of them, they don’t need to. There are horses in most races at odds most punters would shy from, that still have a good chance of finishing in the money, with a bit of luck. And there’s plenty of luck. It’s not going to run out anytime soon.

When we punt we’re gambling; we can’t know the outcome. If we go into a casino and watch people playing roulette for example, many of them are trying to force a favourable outcome to a series of bets, by using staking systems. They reason that certain numbers or combinations are ‘due’ and bet accordingly, citing some undefined “Law of Averages” as the basis for their actions. But the roulette wheel has no memory; every spin is completely independent from the ones before and after it. It’s a mathematical impossibility to ‘force’ a favourable outcome to a series of bets, because it’s a gamble. However, it’s quite possible to win at roulette, if we play for luck, and having achieved some, are content with our winnings and walk away, to come again some other time. If we get greedy though, and want to go on winning for as long as we like, our downfall is assured, because of the House advantage.

It’s the same with horse playing. No matter how smart we are, we can’t eliminate all the luck that works against us and ensure a favourable result. If we play to try to do this, we start following the crowd who are doing the same, and lo and behold, we end up on the favourite (or something very close to it). We need the Probability that something will go wrong for the favourite (which happens just over 2/3 of the time), to work for us, not against us. Fortunately, because we can back for the Place as well as the Win, our choice doesn’t have to have everything go right, just some things, and that will be sufficient for us to move slowly ahead over time.

If we can start looking in areas which maybe don’t get a lot of attention, maybe things which at first seem unlikely, we might find some factors which, while defying our ability to define them logically, nevertheless produce results frequently enough that we can say

“If such-and-such is the case, there’s a Probability of x% that a certain outcome will occur”.

When we’ve done that, we’ve found an edge, especially if the payoff is a greater percentage than what is Probable. Good hunting.

*The Power of Logical Thinking, vos Savant (1996:15)