Whenever we encounter the word probability, I think we should analyze the situation and take it with a grain of salt, for, probability is not always dictated by independent sequence of events, but could be impacted by the history of events as well. The face value of probability may not make sense in a lot of cases though we are misled by our intuition to believe it the other way.

For example, if records indicate that an area is in earthquake zone with a probability of a major earthquake to happen once every 20 years… Forget about how much history went into predict such a probability for a moment. If the last major earthquake happened 19 years before, is it still good to look at it as a 1 in 20 year probability? In reality, a lot of probabilistic situations converge towards certainty as time goes on. In some cases like life span, we are aware of it. But, in a lot of other cases, we just look at numbers and assume it the other way. The next time, when you encounter situations where the probability of that situation happening is very less, you can be convinced that you are probably witnessing the most probable case…:)

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Mad Max

June 1, 2007 at 5:56 pm

@ Mindframes: I think there is some confusion here. When we say there is a probability of say 0.5 that there will be an earthquake every 20 years, it also means that there is a 0.5 likelihood that it will not happen. Even in the 19th year, the probability measure will remain the same and will not be updated to say 0.98.

From your argument, what I understand is as follows. Suppose the last major earthquake occurred in 1990. Given there is a likelihood that every 20 years there will be an earthquake, technically we can expect one in 2010. But does it say with certainty, probability of 1 that there will be earthquake every 20 years? Probably not! Because if this was the case then in 2009, you are looking at the prospect of the probability measure degenerating towards its limit (one in this case).

But I don’t think that is the way we interpret probabilities. As for life span, we know that once born people have to die. Therefore trying to estimate the probability of eventually dying, is meaningless, because it degenerates (event happens with certainty). However what is a more interesting question is when is death likely to occur? Similar variants can also be expressed.

More formally, suppose we have the outcome death (y) and factor (x). Say X implies if death occurs with certainty or not. The joint distribution of X and Y is not well defined because there is no prediction involved. it defies the fundamental principle of estimation theory (because there is nothing to estimate when you know for certain). However suppose your X is something like smoking, then it is interesting to characterize the joint distribution of the two to understand how exogenous (or for that matter endogenous) factors affect the outcome. Therefore the information structure is critical when we talk about probability. In essence there should be an estimation problem for any probability measure to be meaningful.

Hence certainty of probability is paradoxical. For events that are certain to happen, there is no probability because there is nothing to estimate.

Mindframes

June 1, 2007 at 6:48 pm

@mad-max: I am not trying to say that probability is paradoxical.. But, I only talk about the conclusions people tend to make out of it or rather the understanding of it… You are correct in your observation… And yes, it is based on the theory that an event has a probability of happening anytime during a given time period and that the event happening atleast once is very high during that period. In that case, the probability increases with each passing day…

Mad Max

June 1, 2007 at 7:51 pm

@ Mindframes: Hmm again I think the interpretation is not correct here. Probability does not increase/decrease as time passes (the way it is presented here), unless new information comes into the system. If new information comes into the system, it automatically implies that the probability measure so computed will change and a direct comparison with the previously computed probability no longer holds. What say you?

Mindframes

June 1, 2007 at 10:48 pm

not sure if I am stating myself correctly… but, I understand your point and what you say is true…