The main thrust of this work is to treat the initial convective phase of a fluid heated from below as a statistical initial value problem. The advantage of the approach is that it allows a continuous bandwidth of modes to be represented in the initial spectrum. We show that if the initial disturbance field is small and has a sufficiently smooth spectrum, then a natural statistical selection process chooses from the initial disorder a perfectly ordered field of single rolls. The scale of this roll is the scale corresponding to the most critical wave-number obtained from the linear stability problem. We relate this solution to the optimal solution which would be obtained by the upper bound procedures of Howard, Malkus and Busse. Moreover, we show in addition, that if the initial disturbance field is weighted in favour of a particular single roll whose scale is close to critical, the final solution reflects the initial condition providing a certain stability criterion is met. In the two-dimensional case we analyze, this turns out to be the Eckhaus stability condition previously obtained by a discrete multimodal analysis.