This leads us to consider that the probability of the occurrence of an impossible event is 0, and the probability of the occurrence of an event which is certain is 1 and that the probability of the occurrence of a doubtful-yet-possible event is some value between 0 and 1.

It dawns on us, perhaps, that the coin example is too idealistic, too trite.  Concede that there is a finite possibility that the coin will land on its edge - this is the real world.  So we write the probability of occurrence as some very small fraction, fr, over 2. The ratio of the two dominant, but equally likely, events (heads or tails) will be numerically vast compared to the edge probability fr.

Our college students knew they faced a finite risk of having to study, but they were gambling very much with the odds against it.

Intuition tells us that the longer a statistically determined event has failed to happen, the more probable becomes its occurrence. (Fruit machine players believe this).  Conversely, though it may be an absolute rule that in a fruit machine the odds remain fixed each time the sequence starts, by definition, if the odds on a certain improbable event (jackpot) are finite, each such improbable event, as it takes place, is viewed as taking the re-occurrence of the same event further away.  Yet this, on a strict statistical basis, is wrong - it could as easily happen next time round.

Likewise, we might reason conversely, that the failure of a statistically predictable (and normally distributed) event to occur over time brings the likelihood of occurrence closer. We would be very surprised (though it is equally probably) if all the predicted events occurred at the end of time.  But, unlikely as it is, this could happen. Nevertheless, in terms of Murphy’s Law this is the point where dry calculation losses touch with reality.  In the real world sheer statistical probability is just one factor amongst other effects and influences.

Back to our cog (above). The numerical ratios are arguably a reflection on the possibility of unlikely events occurring often. In terms of occurrence, the number of situations in which we find ourselves throughout our lives means that we are continually becoming players in improbable events. In terms of a perception of time and circumstances, very unlikely events appear to happen regularly to some people. Sometimes, we intuit that it is far too often! Just like the coin, we know life can land on its edge and that this remote possibility will happen only when we are least prepared, or equipped, to handle it.

Can we conclude anything from this? 

Only that our lives are not wholly measured by chance, nor are we prone to be the victims of an ill-fated jinx. The more we do, the greater the opportunity for becoming an unwilling participant in a Murphy scenario. Murphy occurrences, and our perception of them, increase in direct proportion to circumstances and our ability not to be able to cope with them - a fact with which we are all acquainted.  Fate rules. We have only limited control over the conditions that optimise the factors governing Murphy’s Law.

So, let us try to formalise Murphy’s Law and give structure to our empirical statement. The possibility of a Murphy occurrence (Mo), through the coincidence of any one of n different events of improbability (p~) able to take place over a given time span (t) is:

Mo=t(n εp~)

(ε =summation of single event p~ terms). Now, Mo approaches 1 as n and t are increased and p~ becomes more finite (increasing to 1).  Time is important because event sequences are invariably time dependent, thus as t becomes vanishingly small Mo vanishes accordingly. Likewise, the absence of an event or event sequence over unit time means that a right-hand term reduces to zero and accordingly Mo is zero.  As p~ increases positively over unit time so too does the value for Mo but this may be countered by a reducing value for t (time span).

This restates Murphy’s Law in a way which denies any supposed demonic intervention, and accounts for the more pragmatic aspects. The message is simple, i.e. in any given time span there is a high risk of the occurrence of highly improbable events - but they may not always be negative and you may not always participate in them.  However, as regards chains of adverse occurrences and unfavourable events, they can and do appear to cluster.

Hence, the explanation is not, as advocated by some, that a large number of improbable outcomes leads to the certainty of an improbable outcome, but that in Murphy’s Law it is the improbable (inexplicable) outcome which tends to be probable!  The two statements are not tautological, since in the first instance, a Murphy occurrence would mean that any observer insisting that an outcome was so uncertain (i.e. certain to be improbable) that the result could not be predicted, would be demonstrably wrong! Likewise, in a field of likely outcomes, Murphy dictates that the outcome is not predictable. 

Murphy will by definition confound. It suggests that the improbable is favoured - in a nutshell there is a high probability that a highly improbable event will occur, or the chances of participating in an improbable event is high.  This doesn’t imply that improbable events are themselves made intrinsically probable, rather their occurrence is.  We distinguish between the implicit improbability of an event in itself, and the likelihood of such events occurring.

As a final comment, some observers will hotly deny Murphy’s Law because it implies bias in the laws of chance - they rightly insist that probability is just that, and cannot be violated, without realising that Murphy’s Law implies no such thing -  since each Murphy event has its own statistical identity.  That is to say, a Murphy is not removed from the statistical environment.

Others avoid acknowledging any contradiction by insisting that we be reminded that adverse events are as equally probable as favourable ones, again misunderstanding the basic Murphy dictum. (Forgetting, for instance that getting it right is one critical path requiring control, getting it wrong is a plethora of pathways independent of control).  Some find refuge in fatalistic acceptance, yet others refuse the evidence of their own experiences and, entirely un-flustered by misfortune, simply ignore it altogether.

Pragmatists on the other hand recognise that it will get you in the end and plan accordingly.