Thursday, January 13, 2005
Even highly unlikely events can occur
Smallholder, over at Naked Villainy, in one of the paragraphs of his post mentions a very unlikely event, namely the recovery of his watch, and wants to attribute it to the assistance of a Supreme Being. This is a very natural impulse and one I have frequently. I tend to be assymetrical about it, only the improbable good stuff gets credit given to God. I usually say S... happens when it is bad.
However, it isn't a bad idea to look at the real probability of finding the watch rather than the perceived one. The conditions to look at are:
a= Probability of capsizing at a particular point
b= probability of being barefoot while white-water rafting
c= probability of a watch remaining in a given location when lost
d= probability of a watch coming to rest at a given location
g= probability of someone reporting finding the watch rather than keeping it
h= probability of the same guide being on both trips
e= probability of the loss being remembered by the guide
f= probability of the guide finding Smallholder's name in the book.
The remainder have probabilities of 1 or almost one.
The overall probability is a*b*c*d*e*f*g*h for one trip, and the overall for a season is the probability of one trip times the number of trips in a season.
Since the raft capsized at that point, I would assume that other rafts would do the same with some frequency. Let us say that a= .25
I would expect b to be very high, and I would count wading slippers almost the same as barefoot lets give b= .8
A watch is quite heavy relative to the water, so that if it falls off it will quickly reach an eddy, sink and being below the current, so once it comes to rest c = 1.
Since one doesn't stand up after capsizing unless the water is calm, the probability of the watch being in a particular eddy is probably proportional to the number of eddies just past a common capsizing point. give d=.05
I would imagine the loss being readily remembered by the guide, especially a female guide. The watch was an engagement present--very romantic, and I would imagine Smallholder made quite an impression with his concern over its loss. (I sure would have) Give e=.99
Having gotten to e, it would be foolish to consider f for less than 1. Even if there were more than one Mark, she would have followed up with all of them. so f=1
The majority of people in this country are honest, and I would suspect that the qualities that select for a desire to whitewater raft would also encourage honesty. I think g=.999
The probability of the same guide depends on the size of the rafting company.
Say there are two guides, so h=.5
we have for a given trip, a=.25,b=.8,c=1,d=.05,e=.99,f=1,g=.999,and h=.5.
the overall probability is .0049 for one trip. Let's say 2 trips a day during the season, which I guess is about a month, so 60 trips, gives a probability of .30, or about 1/3 of the watch being found in a given season. (You can play with the individual probabilities, but when you do, don't calculate the answer till ALL the playing is done. Otherwise you will be guilty of trying to get a certain answer.)
This kind of thinking, that a perceived probability is far less than the actual has a counter in thinking that a perceived negative probability is much higher than actual, e.g. the chance of being on another terrorist hi-jacked plane after 9/11. I was flying the next week, but many people still won't fly. They do not see the thousands of successful flights daily.
It is possible that there was some divine assistance in the return of the watch, but the probabilities would indicate that it wasn't necessary.
[See what happens when you call me 'star-student-of-geometry-class-proof'. It goes to my head. My wife says "Don't encourage him." :-)) ]
However, it isn't a bad idea to look at the real probability of finding the watch rather than the perceived one. The conditions to look at are:
a= Probability of capsizing at a particular point
b= probability of being barefoot while white-water rafting
c= probability of a watch remaining in a given location when lost
d= probability of a watch coming to rest at a given location
g= probability of someone reporting finding the watch rather than keeping it
h= probability of the same guide being on both trips
e= probability of the loss being remembered by the guide
f= probability of the guide finding Smallholder's name in the book.
The remainder have probabilities of 1 or almost one.
The overall probability is a*b*c*d*e*f*g*h for one trip, and the overall for a season is the probability of one trip times the number of trips in a season.
Since the raft capsized at that point, I would assume that other rafts would do the same with some frequency. Let us say that a= .25
I would expect b to be very high, and I would count wading slippers almost the same as barefoot lets give b= .8
A watch is quite heavy relative to the water, so that if it falls off it will quickly reach an eddy, sink and being below the current, so once it comes to rest c = 1.
Since one doesn't stand up after capsizing unless the water is calm, the probability of the watch being in a particular eddy is probably proportional to the number of eddies just past a common capsizing point. give d=.05
I would imagine the loss being readily remembered by the guide, especially a female guide. The watch was an engagement present--very romantic, and I would imagine Smallholder made quite an impression with his concern over its loss. (I sure would have) Give e=.99
Having gotten to e, it would be foolish to consider f for less than 1. Even if there were more than one Mark, she would have followed up with all of them. so f=1
The majority of people in this country are honest, and I would suspect that the qualities that select for a desire to whitewater raft would also encourage honesty. I think g=.999
The probability of the same guide depends on the size of the rafting company.
Say there are two guides, so h=.5
we have for a given trip, a=.25,b=.8,c=1,d=.05,e=.99,f=1,g=.999,and h=.5.
the overall probability is .0049 for one trip. Let's say 2 trips a day during the season, which I guess is about a month, so 60 trips, gives a probability of .30, or about 1/3 of the watch being found in a given season. (You can play with the individual probabilities, but when you do, don't calculate the answer till ALL the playing is done. Otherwise you will be guilty of trying to get a certain answer.)
This kind of thinking, that a perceived probability is far less than the actual has a counter in thinking that a perceived negative probability is much higher than actual, e.g. the chance of being on another terrorist hi-jacked plane after 9/11. I was flying the next week, but many people still won't fly. They do not see the thousands of successful flights daily.
It is possible that there was some divine assistance in the return of the watch, but the probabilities would indicate that it wasn't necessary.
[See what happens when you call me 'star-student-of-geometry-class-proof'. It goes to my head. My wife says "Don't encourage him." :-)) ]
// posted by Bill @ 10:21 PM 
