January 17, 2005

Playing With The Variables

Bill did some nice work with the statistical likelihood that my watch would be recovered. However, I would quibble with his variable a; probability of the raft capsizing at a particular point. Assuming that our raft had capsized at that point, he assumed that rafts capsizing at that point would be fairly frequent and assigned a variable of .25 to the value.

Actually, our raft did not capsize at that point. I must have, while paddling furiously, banged my wrist on the innertube, releasing the lock pin. In fact, we did not capsize at all.

I would wager that very few rafts run into problems that dump out the occupants. The occupants might lose the occasional oar, but they are big floaty things with a wide base. But lets be generous and say that 25% of rafting expeditions experience a catastrophic dunking during their trip. So we'll leave a at .25, but rename it "chance of dunking." The question is, where will the accident happen and how likely is it that it would happen in a particular point - we need a new variable -- aa - likelihood of dunking at the exact location.

Setting aside Aristotle's fox/rabbit "infinitely closing gap," in reality there are a finite number of places that a raft might overturn. Let's assign the area of overturning to be eight feet by eight feet - about the length of the raft. If a raft overturns, the people walking around to right it should step all around this area, and if the location coincides with the resting place of the watch, probably step on it. So, using this arbitrary benchmark, we now need to find the number of locations in which a dunking might occur.

Estimating the length of the trip is difficult. For a two hour trip with a drift rate of, say, 3 miles per hour, a raft would cover 6 miles. So that would be linear 31,680 feet - or 3,982.5 possible locations. Assuming that 90% of the trip is placid floating between the excitement of the rapids, eliminating 90% of the locations gives us 398.25 locations.

Linear locations.

Because rafts don't follow down the river in the same exact path as all previous and future rafts. They meander side to side, limited only by the banks. If the average width of the James River over the length of the rafting route is 300 feet. We might exclude thirty feet on either side of the shoreline; you want to trend towards the middle to avoid running aground. This gives us (300-30-30)/8 width of raft = 30 side to side locations. Further assume that the patterns during the rapids (we have already eliminated the calm portions of the river), funnel the rafts into a few channels. So eliminate 90% of the side-to-side positions.

This gives us 398.25 linear locations and 3 side to side locations, for a total of 1194.25 possible locations for the raft to capsize. The watch is only at one of those locations. So, 1/1194.25 gives us a value for aa of .008367.

Of course, this positional reasoning would eliminate Bill's "d" variable, doubling the chance of the watch being found.

Perhaps I'm guilty of "trying to get a certain answer." I don't think so. This exercise was solely because reading Bill's reasoning had me chewing over the probabilities for the last couple of days.

It is probably irrational to attribute an unlikely occurrence to divine intervention: "God loves me because my watch was recovered." One would be just as justified in attributing an unlikely negative event to divine retribution: "God hates me because I have throat cancer."

Eh, whatever.

Thanks to Bill for providing some interesting intellectual diversion.

UPDATE: Actually, Skippy, God DOES hate you. And Bret Favre. But the rest of us are cool.


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