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 Question from Ben, a student: what does "almost surely" mean in probability? what's the diff between absolutely certain and "almost surely"? thx.

Hi Ben,

According to Wikipedia

In probability theory, an event happens almost surely if it happens with probability one.

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

Probability(the number you roll is larger than 4) = 2/6 = 1/3

Notice that for any event

0 ≤ probability of the event ≤ 1

The difficulty arises if the sample space is infinite. Consider this question

(a) Choose a positive integer at random. What is the probability that the number you choose is larger than 1?

(b) Choose a positive integer at random. What is the probability that the number you choose is less than or equal to 1?

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

Choose a positive integer at random from {1, 2, 3, ..., 100}. What is the probability that the number you choose is less than or equal to 1?

The probability is 1/100 = 0.01 = 10-2. But p is smaller since the sample space in (b) is larger. What about?

Choose a positive integer at random from {1, 2, 3, ..., 1000}. What is the probability that the number you choose is less than or equal to 1?

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,
Penny

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