This is one of my favourite optical illusions:

You can also see it on Richard Gregory’s website and read about it in his last book.

Like optical illusions, wrong beliefs can be impervious to the hardest evidence. That is why many people believe weird things, despite the evidence to the contrary is as hard as it gets. For example, you may have reasons to believe – as homeopaths do – that ‘specially treated’ water is an effective remedy against a particular disease. This is fine, as long as you can show you are right. To do so, all you need to do is to take a random sample of patients suffering from that disease, give half of them a pill soaked in your special water and give the other half a sugar pill (placebo), in your chosen dosage and duration. Neither you nor the patients should know who gets which pill until the end of the treatment. At that time, count how many patients have been restored to health (however defined). Then, among them, find out how many have been treated with the special pill and how many with the placebo. For the special pill to be considered effective, you need to show that most of the recovered patients have been treated with the special pill rather than the placebo. As an example, in Table 1 below, 10% of patients have recovered. Of those, 1,900 have taken the special pill and 100 have taken the placebo. As a result, you can say that the probability of recovery after taking the special pill is 1900/10000=19%, much higher than the 100/10000=1% probability of recovery after taking the placebo. The special pill is highly effective.

Table 1 Effective ‘Special pill’ trial results

Now compare Table 1 with Table 2. The recovery rate is still 10%, but in Table 2 half of the recovered patients have been treated with the special pill and half with the placebo. As a result, the probability of recovery after taking the special pill is 10%, the same as the probability of recovery after taking the placebo – and the same as the recovery rate. The special pill is completely useless.

Table 2 Ineffective ‘Special pill’ trial results

Your belief about the effectiveness of the special pill should be entirely dependent on whether trial results more look like Table 1 or Table 2. And if they look like Table 2, you should abandon your belief. But homeopaths refuse to do so, despite the fact that trial results on homeopathic medicines show, repeatedly and inexorably, that their effectiveness is indistinguishable from placebo. Using our notation, in Table 1 the Base Rate is 10%, the True Positive Rate – or, as it is known in clinical trials, the Sensitivity – is 95%, and the True Negative Rate – or Specificity – is 55%. Hence the Posterior Probability is 19% – almost twice the Base Rate. In Table 2, the Base Rate is also 10%, but TPR and TNR are 50%. Hence the Posterior Probability is 10% – the same as the Base Rate. The useless pill in Table 2 is worth as much as a coin toss. What keeps the homeopathic credo alive is the same phenomenon that gives credence to useless tests and worthless experts: the Prior Indifference Fallacy. Homeopaths look at the 1,000 patients in Table 2 who recovered after taking the special pill and say: “This is tough disease. Only 10% of patients recover at all. But our pill cured half of them: not bad!” The fact that the other half recovered after taking the placebo fails to dampen their enthusiasm. But the biggest boost to the homeopathic delusion comes from high recovery diseases. A common cold, for example, has, given enough time, a 100% recovery rate. This means that the probability of recovery after taking a homeopathic treatment is 100%. Enthusiastic homeopaths will gloat on this piece of hard evidence, neglecting the fact that placebo – as well as dressing up as Elvis – will have the same effect.