It is hard for people to get their minds around small probabilities, very long times, the very small, and the very large, which is why people have a hard time accepting the theories of evolution, quantum mechanics, relativity, and cosmology, which depend crucially on them.

In the case of evolution, the combination of very low probabilities with very long times gives such surprising results that some people are skeptical about evolution being able to create life in all its diversity. Yet, they are unwilling or unable to make the effort to actually do the math and see for themselves. That is fine. We can’t all do the grunt work necessary to convince ourselves of everything. But to then use incredulity as if it were an argument is not a good thing.

Probability is another thing that very few of us have an intuitive feel for. And the intuitions that we do have can lead us astray. This is why people attach great significance to events like someone dying after they dreamed about them or hearing from someone they had casually thought about a day earlier. It seems so unlikely that they think it must have some causal connection or deeper meaning. In reality, what would be truly extraordinary is if people experienced no such coincidences in their lives.

I was thinking about this after I posted about the rise of mega-preachers in Africa who claim to be able to produce miracles. One such preacher who has reportedly 500,000 people attend his monthly services tells a skeptical reporter that people would not flock to his church in large numbers if there were no miracles. On the surface, this seems like a reasonable argument. So if we exclude false information or fraud, what might be happening?

I suspect that the simple answer is probability and the loose way that people use the word ‘miracle’. To me as a scientist, a miracle is a violation of the laws of nature. What determines if something is a miracle can depend on both the nature of the event and its reproducibility. Something may be such a blatant violation of everything we know about how the world works, that it would qualify for a miracle, even it happened only once. Others, if they are merely low probability events, require repeatability to convince me that they are not merely an outlier caused due to the laws of chance. So if an amputee’s limb suddenly grows back to be fully functional, it would have a serious claim to be a miracle. But if someone’s cancer went into remission as a result of prayer, I would want to see reproducibility because cancers can and do go into remission spontaneously because the human body is a complex organism whose workings we do not fully understand.

But many people have much looser standards for what constitutes a miracle. For example, if someone has been out of work for a long time and then gets a job offer, that would be welcome news but not a miracle. However, if that person had happened to make a special prayer just before the job offer came, that same job offer may now be seen as a miracle, an answer to her prayers.

So let us say that the 500,000 people come to church and pray for something special. Let us assume that it is for something like getting a job or passing an exam or getting some sorely needed money or recovering from some illness or finding love or resolving some personal conflict in their lives. Suppose the probability of that happening purely by chance within a week of praying at one of these services is 0.001% (or one chance in 100,000). This is rare enough that if it happens, a gullible religious person would likely consider it a miracle, a direct response to prayer.

What are the chances that at least one person in the congregation will experience such a miracle? It is not a hard calculation. There is a 99.999% chance (probability 0.99999) that for any given person nothing will happen in response to their prayers within the next week. However the probability that nothing will happen for all the half million people is 0.99999^500,000. This number is actually very small, equal to 0.0067. In other words, the chance of no miracles happening to anyone in the congregation is only about 0.67%, or alternatively there is over 99% chance that there will be at least one miracle the following week. In fact, there is a good chance that several such miracles will occur.

Given that people who experience such an event are likely to be overflowing with gratitude (“It’s a miracle! Praise the Lord!”) and broadcast their good fortune far and wide, stories of such miracles will spread like wildfire. No wonder people believe that these mega-preachers can deliver the goods. In fact, once the size of your following gets large enough, you are assured of such miracles routinely occurring, which explains the success of televangelists who confidently promise that good things will happen to those viewers who give money to them, and can truthfully tell stories of people whose prayers were answered thanks to their intervention.

Thus thanks to the general ignorance of probability, anyone can become a miracle worker. All you have to do is plant the seeds of belief in enough people to enable you to take credit for random chance events.

The radio program Radiolab repeated on April 26, 2013 its program on stochasticity that first aired on June 15, 2009. It is an hour-long program but the first 21-minute segment gives you a pretty good idea of how you can be easily misled to see ‘miracles’ where none exist.