Control group much much larger than test group I'm a complete noobie in statistics and this is just me being curious about a study I read online.  In the study they had 150 people who had tested positive for disease X, and a control group of 4000 people who had tested negative.  In the test group 7 people developed disease Z, and in the control group only 3.  So about 5% in the test group, and around 0.08% in the control group.  The study then went to say that the odds multiplier was 60, and therefore people with disease X are sixty times more likely to develop Z than people without disease X.
It seems to me that the test sample size is really low and the control sample size is incredibly high. It doesn't seem very far fetched to think that you could take 2000 people and still get those 3 that developed Z, and then it would be thirty times more likely.  Or take 1500 people rather than 150 for the test group and get 40 people with Z, reducing the percentage to 2.5%.
Is a study done with those kind of control and test group sizes reliable in its prediction?

First off, you should always cite other people's work. Why ask a question about a study and not link to it? We are curious too.
As is, your description omits important information: how was the data collected and how did the researchers analyzed it. Nevertheless, I'll make a couple of conjectures.
You find the study unconvincing because the test group had only 150 subjects with disease X. Perhaps disease X is rare and it would have been hard to recruit more participants in a reasonable span of time. In any case, it behooves scientists to learn whether disease X increases the risk for disease Z sooner rather than later, so that patients can get better & timelier treatment.
Notice by the way that this is not a randomized control trial (RCT) because researchers cannot assign subjects to the test or control group randomly. It's more likely a prospective study: they recruited as many patients with disease X as they could (one cannot simply "take" people for their study), and yes, a large number of participants who didn't have disease X (that would have been easier). Then they waited to see who developed disease Z and who didn't.
Finally, it's of course easier to estimate the risk of developing disease Z from a sample of size 1500 than a sample of size 150. (In statistics by "easier" we mean with smaller standard error.) However, if would have taken the researchers, say, 3 more years to recruit 1350 more patients with disease X, that probably wouldn't have been wiser or more practical.
