20180722, 20:39  #34 
Einyen
Dec 2003
Denmark
2933_{10} Posts 
Why can an infinite series then not have a converging sum below 1 ?
Consider an infinite number of events. The first event have 0.1 or 10% chance of success, 2nd one have 0.01 or 1% chance of success, 3rd one have 0.001 or 0.1% chance of success and so on. Can we agree that the combined chance of any success is 0.11111111..... which is 11.111111....% chance? The chance of Fermat primes falls much faster than a factor of 10 each time, so the combined sum will be much smaller than 0.1111111.... 
20180722, 21:06  #35 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2^{3}×3×79 Posts 
I just don't think any probabilities other than 0 and 1 make any sense in infinite trials.
Think about it what would be the probability of success in infinite trials if the sum of probabilities of individual finite ranges added to a number greater than 1. Surely the probability could not be greater than 1. A fractional probability in infinite trials is as nonsense as a probability of greater than 1 or greater than 100%. 
20180722, 21:57  #36 
Aug 2006
31×191 Posts 
No. If the probability was constant and greater than 0 that result would obtain, but without that condition you don't get the result. Imagine you buy a megamillions lottery ticket on day 1, two on day 2, three on day 3, and so on. What are the chances that on some day all of your tickets will win? Very low indeed! In fact, nearly the entire chance rests on the chance that you win on day one, because the chance of winning with 2 out of 2 tickets is tiny and it just gets worse from there, and even going out to infinity doesn't help (as can be shown with some calculus).
Last fiddled with by CRGreathouse on 20180722 at 21:59 
20180722, 21:58  #37  
"Curtis"
Feb 2005
Riverside, CA
19·227 Posts 
Quote:


20180722, 22:09  #38 
"Rashid Naimi"
Oct 2015
Remote to Here/There
11101101000_{2} Posts 
Perhaps this would be a more relevant example.
Suppose the probability of success at each of n trials is: (19/20)^n So each trial has a probability of success that is greater than 0 and more importantly less than 1. The sum of infinite series 19/20+(19/20)^2+....=19 What is the probability of success in 19 trials? What is the probability of success in infinite trials? 100% or 1900%? Last fiddled with by a1call on 20180722 at 22:11 
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