Should I tell my supervisor about a "critical fail" that I did or do I just let it be and pray that it doesn't get noticed?

Not really sure if this sub is the best place for this, but I'm very new at my job (a month of out of training) and don't have a trusted colleague to ask, so here goes.

Please note that I'm not from the West and English is not my first language, so I hope the terms that I'm gonna use aren't too foreign. If you want me to clarify anything then I'd be glad to.

Part of my work is email support for our brand. I made a mistake, that's not financially impacting to the customer or anything, it's just that they won't receive the email sent which was basically just a confirmation of what he wanted.

QA is not really a part of our metrics but still part of the business. What I did falls under a critical fail by their standards and may merit a final written warning (the next instance being suspension then termination). This isn't a huge deal usually and it has a warming period of a few months, but since I'm on a probationary period at work, it will be considered on our 5th month review and can possibly lead to termination (non-regularization).

QA randomly audits cases, and they usually only pick a few. Others can see the email cases you've handled but it's really up to them whether they'll report the mistake you've made or not.

So I'm torn between reporting what I did to my supervisor so I can at least rectify it (on the business side), and just wishing that no one else notices. If I report then my supervisor may be compelled to report it and I get served the warning, but I'm thinking that at least she'll see me as someone who's willing to own up to my mistakes and do something about it. If I just shut up I maybe able to get away with it, but if I won't then I'll get served the papers anyway but my supervisor will see me as someone who just makes mistakes.

Maybe I'm just overthinking this. I really like and need this job so I'm so pissed at myself. If anyone can point me the right way, I'd greatly appreciate it.

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