I think you are misunderstanding the question.1000 numbers are generated. If you take every four numbers as a sequence then you have 250 possible sequences where you could have four even numbers. There are actually 997 potential sequences, but the 250 is a subset of the total that is easier to work with. So if the odds are minuscule that none of the 250 groups are all even, then it means that the odds are even more minuscule that none of the 997 groups are all even.
Here's the question repeated:
My interpretation: we have 1000 random samples. Each sample can have a numeric value between 1 and 1000. When we say "four consecutive numbers" we intend to mean "four numbers in order of which they were generated" and not "four numbers in numeric order from least to greatest or greatest to least".2) 1000 random integers are generated randomly with a uniform distribution over the range 1 to 1000 inclusive. Which of the following would indicate a poor generator?
(a) the average of the numbers is about 499
(b) each number appears exactly once
(c) no four consecutive numbers are all even
(d) two of the above
(e) all of (a), (b), and (c)
Every time a new random sample is generated, we look at the last 4 random samples. Are they all even? I would hope not! Sure, it might happen every once in awhile, but the chances are low. Hence, the wording of the answer:
"NO four consecutive numbers are all even"
Having no four consecutive numbers that are all even would be a good indicator that my random number generator is truly random. Having 1 or 2 sequences that are all even might even be fine as well. Having several sequences that are all even would be very very bad.