Аннотация: The probability of achieving a goal depends on how the task is broken down into stages and in what order these stages are implemented. This article is devoted to analyzing the optimal selection of the sequence of steps toward the goal.
Introduction
In classical probability theory, the conditions of a problem are fixed, and the probability of an event is determined once. In entrepreneurial practice, it's different: tasks are repeatedly refined, plan stages are rearranged, deadlines and resources are revised. As a result, the probability of achieving goals changes along with the sequence of actions.
Lemma. The probability of achieving a goal depends on how the task is broken down into stages and in what order these stages are implemented.
Illustrative example for proof by contradiction (transportation problem).
Suppose it's required to get from A to C via B, starting at 9:20. AB=BC (distances are measured in straight lines, and movement occurs along the same for simplicity).
Car (minibus): departs every 1 hr 25 min (starting at 9:00), travel time between two adjacent points is 10 hr 10 min.
Airplane: departs every 45 min (starting at 9:00), travel time between two adjacent points is 50 min.
Goal: arrive before 21:55.
Option 1. Car → Airplane. Arrival at 21:50, success (probability = 1).
Conclusion: the order of stages directly affects the probability of success.
Formalization.
Let the project consist of stages i=1,...,n.
Each stage is characterized by a duration ti and a conditional probability pi.
Total probability then:
Psucc=∏(from i=1 to n)pi,
log(Psucc)=∑(from i=1 to n)log(pi).
Discrete case.
Optimal stage ordering (with time constraint ∑ti(C) = < T):
sort in descending order of (Δlog(pi))/ti.
This is analogous to Smith's rule (WSPT) in scheduling theory.
Continuous limit.
We break the process into micro-stages of length dt. Introduce:
g(u,t) - instantaneous increase in log-probability of success,
h(u,t) - failure risk intensity.
Probability dynamics:
d/dt(ln(S(t)))=g(u,t)−h(u,t),
S(T)=exp(∫(from 0 to T)(g(u,τ)−h(u,τ))dτ).
Selection rule: at each moment in time, perform the action that maximizes
g(u,t)−h(u,t).
Thus, the optimal strategy prioritizes actions that:
- provide the greatest increase in probability of success per unit of time;
- reduce the risks of future failures;
- increase the success probability of subsequent steps.
Graphs: the figure shows how the 'success gain' g(t) decreases over time, the risk h(t) fluctuates, and the balance g(t)−h(t) indicates zones where taking action is beneficial (above zero) and where it becomes risky (below zero).
Practical conclusions:
- stages with high probability of success or with an 'unlocking' effect should be performed first;
- the finer the breakdown into stages, the more flexible the probability management.
The order should be dynamically revised as conditions change.
This approach aligns with project management theories, operations planning, and survival analysis.
Conclusion.
Step-by-step problem solving for an entrepreneur is a tool for managing probabilities. In the limit of infinitesimally small stages, a simple rule emerges: at each moment, choose the action that maximizes success gain and minimizes risk.
Main recommendation:
At the beginning of the path, choose steps that either have the highest probability of success or increase the chances of subsequent stages.
References
1) Smith, W.E. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3(1), 59-66.
2) Pinedo, M. (2016). Scheduling: Theory, Algorithms, and Systems. Springer.
3) Cox, D.R. (1972). Regression models and life tables. Journal of the Royal Statistical Society: Series B (Methodological), 34(2), 187-220.
4) Klein, J.P., & Moeschberger, M.L. (2003). Survival Analysis: Techniques for Censored and Truncated Data. Springer.
5) Branke, J., & Schmidt, C. (2003). On the influence of uncertainty on scheduling problems. Journal of Scheduling, 6(3), 273-289.
6) Gittins, J.C. (1979). Bandit processes and dynamic allocation indices. Journal of the Royal Statistical Society. Series B (Methodological), 41(2), 148-177.
