The Timeform Knowledge: Basic Probability

It should be clear from some of what has gone before in The Timeform Knowledge that the good bettor should think in terms of probabilities rather than in terms of certainties.

There is only one outcome to an event, but, in theory, if you repeated that event many times you would get a multitude of outcomes. The outcome observed is just one of these.

As every event (such as a race or a performance) will have a subtly different context, it is only by striking bets a large number of times that we may approximate the effectiveness of an overall betting approach.

Thinking in terms of probabilities is more important to a punter than knowing probability theory in detail. But a basic grasp of probability theory will not hinder and may well help in certain contexts.

It is easiest to illustrate the basics of probability by first reducing the number of possible outcomes. The number of possible outcomes of a horse race is almost infinite if you include everything that might happen in that race and not just the order at the finish. The number of individual outcomes of tossing a coin or throwing a dice is finite.

If you want to throw a six with a dice (assuming a fair dice and fair environment) then the probability of this occurring is defined as the number of favourable outcomes divided by all possible outcomes, or 1/6.

In the language of probability, this would be expressed – on a scale from 0 (no chance) to 1 (certainty) – as 0.1666 recurring.

The probability of an even number (2, 4 or 6) occurring would be 3/6, or 0.5, while the probability of ANY number showing, given that we throw a fair dice, is 1 (certainty) in this context.

It follows that the probability of any number OTHER THAN six showing is this second probability (1, or certainty) minus the probability of throwing a six. That is, 1 minus 0.1666, or 0.8333 recurring. This is known as a “complementary event” in probability.

Events may be independent, rather than complementary, such as the probability of throwing two consecutive sixes, again assuming a fair dice and true independence. The calculation of this is 1/6 multiplied by 1/6 = 1/36 or 0.0277 recurring in terms of probabilities.

One aspect of probability which often trips up the novice (and even sometimes the expert) is that the probability of throwing two consecutive identical numbers is NOT the same as 0.0277.

It is a certainty that a number – any number, not a specified number – is thrown with the first dice, so it simply becomes that (probability = 1) multiplied by 1/6 (probability = 0.1666) that two unspecified numbers are thrown consecutively. That is, the probability is 0.1666 again.

This can be illustrated in horse racing terms by reference to the US Triple Crown, which is the achievement of winning the Kentucky Derby, Preakness Stakes and Belmont Stakes with the same horse. American Pharoah, in 2015, was the first horse to manage this feat since Affirmed in 1978: a magnificent effort by the horse himself but nowhere near as improbable an event as stated in some quarters. 

The probability of this happening in any given year was not the multiplied probability (perhaps derived from race-day odds) of a given horse in the three separate legs, let alone the implied probability that one specific horse from the tens of thousands bred each year would pull off the feat.

That a horse – any horse – would win the first leg, The Kentucky Derby, was a certainty (for these purposes), so the true probability of a horse winning the Triple Crown was the probability that a horse which HAD WON the Kentucky Derby then won the Preakness Stakes and the Belmont Stakes.

Given that a horse which won the Kentucky Derby would, barring injury, almost certainly contest the second leg, the Preakness Stakes, at a short price, and then, if it won that second leg, contest the third leg as a dual classic winner at a very short price, the probability clearly becomes much less.

Perhaps something like 1 (certainty) multiplied by 0.28 (approximately 5/2 in terms of odds) multiplied by 0.4 (6/4 in terms of odds), which is 0.112, or around 8/1 (though others would doubtless assign different probabilities).

You might expect the US Triple Crown to be won about once a decade. It has been won 12 times in the last 100 years and the “famine” between 1978 and 2015 (which included several near misses) can be seen as unrepresentative.

Moving from the throwing of dice (an “aleatory” activity for those who like to know these things) to the uncertainty and subjectivity of horse racing has taken us into the far more interesting world of subjective and conditional probabilities.

In the above example, the probability of a horse winning the Preakness given that it had won the Kentucky Derby might be 0.28, and the probability of a horse winning the Belmont given that it had won the Kentucky Derby might be similar. But the probability of a horse winning the Belmont given that it had won the Kentucky Derby AND the Preakness will nearly always be greater.

The last-named scenario includes additional information, with probabilities clearly being influenced by the hypothetical chain of events. That is why bookmakers quote separate odds about conditional or contingent events, such as a horse winning both the 2000 Guineas and The Derby.

Probability Theory – not to mention common sense – shows that they are right to do so.