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Offlinedaba
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"Once in a while" and "Every once in a while" are they the same or not? Prove it!
    #5841374 - 07/10/06 03:46 AM (17 years, 6 months ago)

So I was having a conversation with my friend, and the phrase "every once in a while" popped up. Then I thought, hmm... what's the mathematical interpretation of this statement, and its sister statement "once in a while"? Are they equivalent? If not, then where do they differentiate?

Anyways, I thought this was pretty interesting. You may need a little bit of knowledge of basic stochastic processes to understand what I'm saying. But... consider the two phrases:

Statement 1: "I go drinking every once in a while."
Statement 2: "I go drinking once a while."

Now lets study how to mathematically model this. Let the event A be "the opportunity arises to go drinking". Then define a Poisson process P(L) with rate L defined to indicate some measure of infrequency. Then the event A occurs according to P(L).

To first study Statement 1, this is simply P(L). Meaning, every time event A occurs, you go drinking. That is, every time the opportunity arises, you go drinking. Simple!

Now study Statement 2, this is an application of Poisson thinning. That is, every time event A occurs, then you have probability p of going drinking and therefore probability 1-p to not go drinking. Therefore, the probability of you going drinking is a Poisson process with rate L*p.

So what can we conclude? That Statement 1 is not the same as Statement 2! The forementioned was a lax proof of it!

I hope you enjoyed it as much as my chat-mate and I did. :smile:


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