



 
Hi Ben, According to Wikipedia
The confusion between absolutely certain and almost surely can arise when the sample space is infinite so let's make sure we are using the same terms. In an experiment the collection of all possible outcomes is called the sample space. Thus if you roll a six sided die and read the number on top the sample space is {1, 2, 3, 4, 5, 6}. An event is a subset of the sample space, so for the example of rolling a die an event is that the outcome is larger than 4. Thus this event is {5, 6}. The probability of this event is a fraction where the denominator is the number of elements in the sample space (in our example 6) and the numerator is the number of elements in the event (in our example 2). Thus in our situation this gives
Notice that for any event
The difficulty arises if the sample space is infinite. Consider this question
I am going to ask instead,
Suppose the answer to (b) is p then the answer to (a) is 1  p. I don't know how to solve (b) since the sample space is infinite but I can answer this question
The probability is 1/100 = 0.01 = 10^{2}. But p is smaller since the sample space in (b) is larger. What about?
The probability is 1/1000 = 0.001 = 10^{3}. But p is smaller since the sample space in (b) is larger. I hope now you can see where I am going, p is less than, 10^{2}, and 10^{3}, and 10^{4}, ..., and 10^{k} for any positive integer k. But I know that p is a probability so 0 ≤ p. So what number is greater than or equal to 0 and less than 10^{k} for all positive integers k? The only possibility is p = 0 and hence the answer to (a) is 1. Thus if you choose a positive integer at random it is almost surely greater than 1 although is not absolutely certainly greater than 1. I hope this helps,  


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