Averages

Probabilities are often used to calculate expected average outcomes. Here is an example. Suppose that a farm can plant either of two crops, S (sorghum) or M (maize). Crop S does best if rainfall is in the growing season is between 300 and 400 mm total. In that range, the farm could expect to earn about $120 per hectare by planting S. With rainfall only between 200 and 300 mm, the crop will be smaller and the farm will earn about $100 per hectare. With higher rainfall, there is apt to be crop damage, and again, the earnings may be about $100 per hectare. Crop M, on the other hand, does best with high rainfall and will be lost entirely in a dry season.

Suppose that the rainfall probabilities are estimated as in the first row of the following table (which also shows likely earnings per hectare from S and from M).

If the farm always plants S, then over a 10 year period it can expect to earn $100 in each of 4 dry seasons, $120 in each of 5 "good" seasons and $100 in the 1 wet season, for a total of $1100, or an average of $110 earned per season per hectare planted. Similarly, if the farm always plants M, then over a 10 year period it can expect to have a total loss on 4 years, but will earn $900 in the other years, for an average of $90 earned per season per hectare planted. Since this average result will be better by planting S, and since there is much less risk in planting S, the farm will most likely plant S every year. Of course, if we could predict the rainfall in advance with perfect accuracy, it would make sense to plant S only in the dry years and M in the other years, leading to an average of $130 per season per hectare. In the absence of any ability to predict, but knowing the historical average probabilities of the three types of seasons, it is clearly better to plant S all the time.

Next, suppose that a WELL-VALIDATED forecast for the coming season estimates 20% probability of rainfall 200-300mm, 50% in the range 300-400mm, and 30% over 400mm. The above table changes (in its first row only):

If the farm continues to plant S in seasons with such a forecast, it will continue to average $110 per season per hectare. However, if it plants M in all years with such a forecast, the average will up to $130 per season per hectare. This is a nice improvement in average outcome, PROVIDED that the farm can survive the expected 2 seasons out of every 10 (20% probability) in which the M crop is a total loss.

The calculation of average return, using probabilities, can be a good way to make decisions (such as planting S or M) PROVIDED that the probabilities are valid (correctly indicate relative frequencies of the different sorts of events, such as wet or dry season) and provided that one does not assume to much risk (e.g. total loss of crop) for the occasions when the less probable even does happen.