Probability Theory Predicts That Group Survival May Be Guaranteed for Groups with More Than 10 Elements

Abstract

The purpose of this study is to find out the critical number of elements needed for group survival. Taking a probabilistic approach, how the lifetime of a group consisting of several elements depends on the number of elements and the probability distribution of their lifetimes is investigated. Four probability distributions are examined: an exponential distribution, a uniform distribution, a parabolic distribution, and a pointed distribution composed of two parabolas. The lifetime of the group is defined as the expected value of the maximum lifetime of the elements in that group. The probability distribution of this maximum shifts to the right as the number of elements increases, and the expected value of the lifetime of each element eventually becomes less than the lower limit of this distribution. The number of elements in this case is defined as the critical number of elements needed for group survival. Hence, if the number of elements is larger than the critical number needed for group survival, the lifetime of the group is guaranteed to be longer than the expected lifetime of one element. The findings are in the following. The critical number needed for group survival is inversely proportional to the expected lifetime of one element, regardless of the probability distribution used. It decreases as for an exponential distribution, and as in case of the others, where N is the number of elements of the group. Because a probability distribution defined over a finite range is assumed to be reasonable in practice, a group consisting of more than 10 elements should survive well.