What are the Odds?

Murphy's Law is an oft quoted assertion to the effect that "anything that can go wrong, will go wrong". Clearly this is not strictly true, as an unlimited propensity of events to conspire against us would render life quite impossible. Seeking a more exact definition, I have attempted to define Murphy's Law mathematically, based on relative probabilities for given events, as a function of their 'undesirability':

The sum of the probabilities of all possible events from a given initial state is equal to ‘1’ (certainty), ie. it is certain that one of all the possible outcomes will occur. Let each possible event have a probability ‘P_n’, then the sum of all possible events: P₁ + P₂ + P₃ + P₄… = [SUM]P_n = 1.

Experience tells us that the probability of a given event occurring is to some degree directly proportional to the annoyance/inconvenience to the observer that is brought about by said occurrence.

However, experience also tells us that there are practical limits to the extent to which Murphy’s Law distorts the nominal probability of any given event:

For example, dropped toast may land butter-side down more frequently than it should- a moderate skewing of nominal probability, but the air in a room never conspires to spontaneously evacuate it- such an event would constitute a much greater distortion than Murphy’s Law brings about.

For a given event of probability ‘P’ then, we can conceive of a maximum coefficient by which the nominal probability can be altered. Let this probability be ‘M’ the Murphy’s Law Coefficient.

Now, for a simple system with two possible events resulting from a given initial state, P₁ and P₂, P₁ + P₂ = 1. Assume that the possible outcome ‘P₁’ is disfavoured by the observer, while outcome ‘P₂’ is favoured:

Incorporating Murphy’s Law, the probability ‘P₁’ then becomes P₁(1+N₁M), where ‘N’ is a number between -1 and +1, representing the annoyance/inconvenience inflicted upon the observer as a consequence of the event occurring. (In the event that an outcome is favoured by the observer, N is less than '0'). The sum of 'N's for all possible outcomes for a given event is necessarily '0': [SUM]N_n = 0.

In the ‘two possible outcome’ case then, P₁(1 + N₁M) + P₂(1 + N₂M) = 1.

Extending this reasoning to the general case then gives:

[SUM]P_n(1 + N_nM) = 1 ([SUM]N_n = 0).

What is the value of the Murphy’s Law Coefficient, ‘M’? By the conventions we employ here, and assuming Murphy’s Law always to have an adverse effect, ‘M’ must be greater than ‘0’. Our experience of the limited effect of Murphy’s Law suggests that ‘M’ must be significantly less than ‘1’.

However, any experiment intended to quantify the ‘Murphy’s Law’ coefficient would inevitably be thwarted by the influence of Murphy’s Law itself.