There are times when two or more parties disagree over when a particular event will happen and the disagreement is so strong that people are willing to bet each other. It is common to place “over-under” bets — if the event happens before time T, Andrew gets the money, otherwise Bob gets it. Usually the further the event is from T, the more money exchange hands.

I don’t like this style of betting because it’s not expressive enough. Instead, I prefer to bet by specifying my probability distribution of the timing of the event, and then using these distributions to determine payouts. It's a fun activity, requiring very little mathematics to execute well.

Essentially, each party graphs a probability distribution of the timing of the event — a histogram with the time on the horizontal axis and the probability density function on the vertical axis. The probability density function is simply “the relative probability that the event will happen around the time specified on the horizontal axis”. So if the histogram is twice as tall around 8pm than around 7pm, the event is twice as likely to happen around 8pm than around 7pm.

That’s all each person really needs to do. No need to worry about the area of the histogram summing up to 1 since the vertical axis can be scaled up appropriately. The two people should also agree on how money they are willing to bet — say k dollars each.

When the event actually occurs at time T, the two people compare the value of the probability density function (the height of the bar) at time T on their graphs and pay up based on the difference in these values. The height of each bar needs to be scaled appropriately so that the area under each curve adds up to 1 – that way, no player can cheat by making their graph "taller".

To do the scaling well, we need to calculate the area under the graph. A few heuristics that work include:

Limiting oneself to "aliased" curves on graph paper, so that the area is simply the number of squares under the curve
Limiting oneself to piecewise linear curves and doing relatively simple math to compute the area
Scanning the graph and using a graphics editing application to determine the area under the graph (using e.g. the flood fill and histogram tools)