A theorem on the optimal allocation of effort

A limited time budget is to be allocated to several tasks, so as to maximize the probability that a majority of these tasks will be performed correctly. It is shown that in the symmetric linear case, it is optimal to allocate time equally among k of the tasks, where k is at least a majority, but may be more, depending on the actual time available. In particular, time is allocated to all tasks if there is little time available, but to only a majority of the tasks if the available amount of time is  reasonably large: