Using positive integers, think of primes ending in 1 or 3 (you may want to use the technically precise phrase 'whose units digit is 1 or 3'). For example, 21 'ends in' 1, but it is not prime. You must go in order and you are not allowed to ask the number the previous person gave!
I tried doing this myself; it's surprisingly hard! I would never be so foolish as to say 51 is prime... but I said to myself "41... 43... um, what comes next?" It's something about looking at that list of numbers out of context; a significant part of my knowledge about which numbers are prime is apparently just being able to recite the first thirty or forty (up to 150 or so) pretty quickly. ("Recite" is not quite the right word; I tried to actually do this and once I got past 37 I realized that I was not quite working from memory so much as doing trial division.)
Also, I have to share the old joke that 91 is the first number that "looks prime" but isn't. The argument is that multiples of 2 and 5 "look composite" (by looking at their last digit); multiples of 3 "look composite" since people know the fact that a number is divisible by 3 if and only if the sum of its digits is; and (two-digit) multiples of 11 "look composite" because of their repeated digits. Furthermore, squares don't "look prime". So the smallest number that looks prime, but isn't, is 7 times 13, or 91. After that, numbers that look prime but aren't are:
119, 133, 143?, 161, 187, 203, 209?, 217, 221, 247, 253?, 259, 287, 299.
(I put a question mark after multiples of 11 because I'm not sure how to count multiples of 11 which are greater than 100. And of course, this criterion of "looks prime" is subjective; for example, I don't think 217 or 287 look prime, since they can easily be written as 210 + 7 or 280 + 7 and thus their factorizations as 7 × 31 and 7 × 41 are obvious)
so there are 14 "psuedoprimes" in the range from 100 to 300, compared with 32 actual primes in that range. So this test isn't too bad in that range. Of course, eventually it becomes quite bad; it identifies (1/2)(2/3)(4/5)(10/11) = 8/33 of all integers as primes (ignoring the fact that it identifies squares as nonprimes), when the actual density of the primes is zero. In the limit, almost all of the numbers identified by this test as primes are in fact composite.