I really doubt the garbage disposal is really broken, we just had two other major things replaced, what are the odds that another thing would break so quickly?

If they're independent, the odds are exactly the same as if two other major things had not been recently replaced. If they're dependent, the odds are higher. For this decision, you have a lot more evidence about the specific, so your base rate (unless it's incredibly small) doesn't matter, and the specific evidence of brokenness overwhelms it.

A better application is budgeting for next month. Compare how much you're planning to set aside for repairs with how much your wife is. See who's right. Update. Repeat.