given 150 marbles numbered 1 to 150, what are the odds that 2 of the numbers between 1 and 10 will be picked within the first 10 tries.
marbles are picked without replacement.

If that happens, what are the confidence levels that there is some bias in the system and that it isn’t random?
Same questions for the odds of picking 2 numbers between 1 and 5 within the first 10 tries?

Wolfram Alpha can do that for you. It says 11.5%, but check to see this is the right equation. I just copy and pasted it and haven’t thought about it myself.

Hmmm…good question. Lets go with ‘exactly 2’ for no other reason as it seems to be easier to figure. I assume ‘at least 2’ would yield a slightly higher probability.

If the calculation is correct than about 11% of the time we should get this result.

If I did this and got that result how confident can I be that this is not due to random chance, but instead due to a bias or trend?

You have 150 numbered marbles. You draw 10 marbles without replacement and then you look at your collection of 10 marbles to see whether two or more fall within the range 1 to N (with N=5 or 10 in the OP).

For this interpretation, I get the following probabilities via simulation for finding at least two small numbered marbles (numbered 1 to N) upon drawing 10 marbles from an urn with 150 marbles.

For instance, was the number of draws (five or ten) predetermined, or did the fact that you drew low-numbered marbles in quick succession influence the stopping condition? Are there other “unusual” sequences of draws that would have pinged your bias radar, or is this one somehow extra special (e.g., what if you saw two numbers between 141 and 150 instead of between 1 and 10)? How many other random sequences took place beforehand that could have triggered your bias radar but did not?

(…and the cleanest way to get around all such questions, given your new suspicion of bias, is to now discard the past data and run a series of fresh draws of ten marbles, counting the number of times your condition of two or more low numbers is met. Is will be a lot easier (read: possible) to answer statistical questions on that unbiased sample that on the sample(s) that sparked the question.)