It’s ok to be wrong as long as you’re wrong in the right way. There’s no fault in arriving at the wrong answer because the problem was difficult and you drew a reasonable but incorrect conclusion based on the available information. That’s how you learn, and people generally won’t hold it against you. Just figure out what you botched and why and move on.
But if you were wrong because you were sloppy or lazy in your analysis, or you were rushing, or you just didn’t care, that’s a problem. That’s how you earn a bad reputation and stop getting choice assignments and new opportunities. Welcome to copy-and-staple duty.
You’ll be amazed at how often you can be wrong and still be seen as the right person for the job as long as you’re the right kind of wrong. But be wrong the wrong way once and see how long it takes you to live it down. Or better yet, don’t. How? Do good work. Every time. That’s the only right-kind-of-wrong-guarantee insurance I know of.