"There are three kinds of lies: lies, damned lies, and statistics." - Benjamin DisraeliHighly improbable events can often seem miraculous, our minds are not equipped for intense statistical analysis and certainly not for dealing with large numbers. Couple this with the very limited scope in which to view the world and suddenly blind chance can seem fate, divine intervention looks to be the only explanation. Certainly some events are too statistically improbable to happen by chance, and in those intelligence must be sought as an explanation. But given enough time and enough of a sample base, miraculous events can and do happen in every walk of life.
Winning the Lottery
Now this is a statistically improbable event. In a system where 6 of 44 numbers are drawn out, the chance of winning is approximately 1 in 5 billion. Given that in the Australian lottery system each ticket has 12 different combinations printed on it, that chance drops to about 1 in 40 million. So if there were a million tickets bought each week, it would take about 40 weeks for the statistical probability to match the unlikelihood. Run the lottery enough times and there's bound to be a winner somewhere.
Now what would it look like from the winner's perspective? They have won something that is incredibly unlikely, a statistical fluke. The pay-off for that statistical fluke is a huge financial gain and the cost was very little. Buying a ticket each week certainly improves the odds, but not by enough for it to be considered anything other than a statistical fluke. If someone bought a ticket every week for two decades, the chance of winning once is approximately 1 in 40,000. So while it's a far better chance than just buying one ticket once, the difference in likelihood of outcome is practically the same. So the person who wins had an extreme bit of luck, it's nothing more than the inevitability that someone has to win some time. For the extremely fortunate individual this could be taken as a significant event, as something more than blind chance.
This is how in the ordinary course of events, people can lead extraordinary lives. The luck that a few have is inevitable in a large enough population size; the scope of statistical insignificance is wiped out. It's with the global media that improbable events are broadcast into our daily lives. We are exposed on a daily basis thanks to the news, something that almost never happens is being portrayed to us as a regular occurrence. So by extension, winning the lottery would sound a lot more a frequent purely because the sample size is extraordinary. This can be applied further.
On a simplistic level, take rolling the dice. Now guessing one roll is 1 in 6. Guessing two rolls is 1 in 36, three rolls is 1 in 216, 10 rolls is 60,466,776. Now if everyone in the world were to guess the sequence, it's inevitable that people would be able to guess the exact sequence. It would be unlikely that no-one was able to guess it; statistically it should have happened 100 times over. This is why picking a single individual event is a good indication of psychic power. Statistically the improbable does occasionally happen, it's only with consistency that meaning can be derived.
Meaning from consistency
Now given the premise that an individual can experience improbable occurrences given a large enough sample size and time, the role of causality needs to be explored. While there certainly are events that befall one for simply nothing more than being in the right place at the right time, there is also the possibility that the individual is rigging the results. This is the train of thought leading to psychic powers, that there are those who can mentally affect the selected outcomes through divine insights. It can also lead to the belief in a personal God (or guardian angel) is watching over the affected individual.
Having a one-off event of improbable chance is statistically explainable; having repeated bouts of chance would indicate there has to be some underlying cause. Say for instance if someone could correctly predict a shuffled deck of cards in order with no foresight (1 in 8.0658*10^67) there must be something to explain it. The obvious answer is that the deck was rigged, that the person knew in advance what order the cards were in. Without that, there is calls for further study to find out why.
As noted previously, appealing to the paranormal is not an explanation; it explains nothing and just raises further questions. Rather a mechanism of how someone could see into the future needs to be explored. But in order to get to that state, there needs to be some consistent statistical significance. Without the ability to reproduce improbable events, then chance is the only viable explanation. We don't ever get something like 52 cards being picked, instead events with higher probability are sought after. Instead it might be something like the value on the card (1 in 13) or pick the suit on the card (1 in 4). It makes the probability of having hits higher, but it also takes away from the significance of picking an incredibly improbable event.
Take picking the suit on the card. For a deck there the average person should pick 13 of the 52 cards by guessing before any card is dealt. If the predictions are made after each card, the odds would change (i.e. if 51 cards are picked out, the observer should be able to deduce the final card's suit). So there should be scores around 13, but not everyone will fall exactly on 13. There should be people who score 15 or 16, some who score 8 or 9, there may be one who scores a 24. If that individual consistently scores in the 20s while others average out over a long period of time to around chance, then there is something significant to derive from the event. If someone scores 22 one turn then 11 the next, then it falls within the normal course of events.
Consistency is the only way to gain meaning from statistics, it's the foundation of statistical analysis. The problem when looking at the paranormal is the tautological nature of the phenomena; it's never that the phenomena doesn't work, it's that the user wasn't in the right frame of mind. As the saying goes: If someone wants to use statistics to appeal to the credence of a phenomena, then all statistics and their context must be shown. If someone can pick the suit of a card 1 in 2 times as opposed to 1 in 4 consistently, then there is good cause to trumpet that stat. But if it happened once and the rest it averages out to 1 in 4, then there is nothing special to report. The improbable does happen, it can adequately be explained by statistics. The sooner people learn to use stats properly the better.