To me this probability math has a few flaws in its considerations.
1) It ignores the factor of time - a failed voyage doesn't just mean less resources it comes at a high time cost.
2) It ignores the practical resource cost of failure - ship repair charges and dead crew costs impact on overall outcome, especially when you consider a crew death not only costs resources to replace it has an ongoing negative effect whilst that replacement is levelled up.
3) It is all very well in theory, but as with most complex probability/statistical odds theory and reality rarely conform taking thousands and thousands of interactions to trend towards the expected; so for a singular user this model is unlikely to play out correctly. It could work out much better than someone using no merchant BUT it could also end up much worse.
I'll address these points in reverse order:
3. You vastly underestimate the power of the central limit theorem. The variance of the mean result is extremely small after, say, 100 voyages, which is a rather low proportion of the total you will have to do. I can very confidently say this will work out better for you, and no, you do not need thousands and thousands of trials to "trend toward expected." In any case, there is variance with and without merchants, and you could end up worse than expected without a merchant, too. Please do the math before making bogus claims about probability theory.
2. While it's true, I ignored this, the effects are rather limited. While I have only my own data to rely upon, I have never lost a crew member (and if you have lifeboats, you don't even risk them), and have never been charged what I would consider a significant amount for ship repairs. By assuming my own mediocrity, I can only conclude that such serious costs are rare and not a significant factor. This would then only become a factor when you have no marginal benefit from acquiring more resources (aka, you already have all upgrades and more resources than you know what to do with).
1. The time cost is also irrelevant, and in fact doesn't exist. There are two kinds of time cost to consider: in terms of lost resources, and in terms of lost distance. The cost in resource acquisition, however, is already taken into account in the prior calculation. The cost in distance acquisition, while seemingly important, is in fact, not worth considering, because you gain the same distance regardless of whether a mission succeeds or fails.
So none of these objections have any weight. I concede that if you no longer care about resources, there's no benefit to having a merchant on a resources mission, but that's just common sense. In every other case, which is all the time until you're in the Pincers and maxed all upgrades, you should definitely have a merchant on all your missions.