# Essay about A Unique Output Based On Fixed Inputs

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Considering that this is a deterministic approach there will be a unique output based on fixed inputs. In this case the X forces win with a probability of 1. The expected number of survivors for the X forces will be 2.83 and Y forces expected survivors will be zero. The battle is expected to last 62.32 units of time between the X and Y forces.
Question 2
A stochastic model was developed in R and the following values listed in Table 1 were inputted individually into the code and both experiments were simulated 10,000 times. The statistical results of this analysis were used to construct Tables 1-3 and Figures 4 and 5 displayed below.
In table 1, probabilities are displayed for both forces under experiments 1 and 2 for each stochastic Lanchester simulation run. Also, Figures 4 and 5, display the winner force probabilities in the bottom right histograms. The X force is the winner for both experiments, therefore, the larger force of X has a probability of 1. As the forces are increased the Y probability becomes zero. Next is Table 2, which displays the statistics gathered for survivors of each force and experiment.
What is interesting about Table 2 is that as the X forces get larger as does the standard deviation, which indicates that the mean value from the actual mean is great when X forces are larger. Also noted, the means are within their 95 percent confidence intervals and furthermore in experiment 2 the Y forces will have zero survivors whereas in experiment 1 the Y forces are nearly zero but not exactly. The histograms for both X and Y survivors can be seen in the top left and right histograms of Figures 4 and 5. Within the histograms, the probabilities of a number of people surviving can be seen unlike for the ...

... middle of paper ...

...ysis of both experiments align with the analysis conducted in the previous questions. No matter the force size the result is always having the Spanish-Franco win the battle. Therefore, as the force size increases the standard deviation gets larger. Likewise from previous analysis, confidence intervals appear to be narrower for survivors when forces are smaller, but the confidence intervals appear to be wider for battle duration when forces are smaller. Moreover, the analysis of the Lanchester Square Law both theoretical and by validating Lord Nelson’s battle using the stochastic process gives a great example of how forces with a 2:1 ratio increased to a 3:1 ratio dominate the battle. For example, Lord Nelson’s second sub-battle with a 3:1 ratio nearly dominated the battle, whereas, the forces in questions 1 and 2 dominated but not exactly like the Trafalgar plan.

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