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It's even worse. The second Borel–Cantelli lemma shows that if the marginal risk of an event (per time unit) doesn't decrease to zero at a fast enough rate, the
by Mattasher 7y ago
It's even worse. The second Borel–Cantelli lemma shows that if the marginal risk of an event (per time unit) doesn't decrease to zero at a fast enough rate, there's absolute certainty it will occur. (strictly speaking, an infinite number of times in an infinite sequence).
- deleted 7y ago[deleted]
- zaroth 7y agoIsn’t this just the limit of the probability of the event not occurring raised to the n-th power as n approaches infinity going to zero? Seems like awfully low hanging fruit to be known as “the second Borel-Cantelli lemma”. I guess there were some benefits to laying out the groundwork of probability theory in the early 1900s.
- Sharlin 7y agoAt least it’s just a lemma rather than a theorem :)
- Mattasher 7y agoSort of. The result is even stronger though. For one, the probability of the event can be tending to zero. So the thing you are multiplying together (in the negative case) gets closer and closer to one. For another, it states that not only is the event certain to occur, it will occur infinite times. In other words (and frighteningly!), no matter far out you go along the timeline of diminishing probabilities, if the rate of diminishment isn't high enough, infinite occurrences still await you.