The United States Navy has found by experience that the proper amount of food to prepare per man for a mess of N men is not a linear function of N. As this is not in accord with what one Might expect at first sight, it was thought worthwhile to consider the problem from the standpoint of the theory of probability. The Navy Cook Book1 says,
All of the following recipes are intended for 100 men. When the number in any one mess is in excess of this amount, the following reductions are recommended:
300 to 500 men, reduce recipe by 8%
500 to 1,000 men, reduce recipe by 10%
More than 1,000 men, reduce recipe by 12%
In a large mess, considerable saving will result by carefully following the reductions recommended.
The object of this paper is to subject this point to analysis in terms of the theory of probability. It will be shown on three different assumptions that the curve which Shows the proper amount of food per man for a mess of N men should have a fractional decrease from the amount per man for a mess of 100 men which is a function of N of the form where k is a constant. In Fig. 1 is shown a plot of the per cent reduction as a function of the size of the mess in hundreds of men. The jagged curve (a) corresponds to the recommendations given in the Navy Cook Book, while curve (b) corresponds to that given by probability theory assuming that the adjustable parameter k has the value (10.16).
We interpret the effect as due to the statistical fluctuation in the amount of food desired by any particular man from day to day. The theory is carried through for three different assumptions concerning the variation in individual demand of the individual man:
(1) Assume the service perfectly continuous and that the probability that one man will want an amount of food between x and x plus dx is given by where m is the mean amount consumed and a is the dispersion.
(2) Assume the service to be such that a man must always take an integral number of units called “portions” or “servings” and that the probability that he take n servings is given by the Poisson law, where m is the average number of portions per man.
(3) Assume that the service is in portions but that a man has only the right to pass up that course altogether or to take one serving, no more. Let p be the probability that he take the serving.
Of the three assumptions we shall see that probably the last is best in accord with the facts.