People do not necessarily need a reason to believe in something: they just might. It might just be part of what they believe, and there’s nothing much you or they can do about it.
As for biblical inerrancy, it’s important to realize that even this minority opinion has many subsets, from god inspiring things word for word to god ensuring the fidelity of the message to the text being written by honest men (ensured honest by god) but in their own words relating what god supposedly related to them (sometimes years later). All these views have subtle implications, and you can’t examine them all as if they were exactly the same.
As for Bible contradictions: there is really no way to convince someone that doesn’t want to be convinced that there is any contradiction. Why? Because by presenting the contradiction, you are dealing with a reversed burden of proof: trying to prove that there is no way that two passages could be anything but a contradiction. But there is simply no way to cover all the bases, and all an inerrantist need do is suggest some additional facts, even ad hoc, that account for the apparent contradiction with extended context. Put plainly: what we have is a negative proof, which is nigh impossible to nail down in practice.
Cliff Walker has examined this extensively with the fig tree example. The basic question is: Why does Jesus curse the fig tree for not bearing fruit when the text of Mark itself says that it was not the season for fruit.
In response to this challenge there is often first the objection that fig trees flowered early that year, and when challenged about this, there are tons of complex accounts attempting to show that the fig tree species was different than we modern people expect, and bore fruit earlier, that it was unseasonably warm, etc. etc. All this, of course, is totally beside the point: no matter what account of fig trees is given, it doesn’t change the fact that the text of Mark, the inerrant text, says explicitly that it was not the time for figs. So while these explanations are totally off-the-wall (and perhaps unecessary, since it isn’t a contradiction per se anyway), the example does demonstrate the sort of complex MO people often undertake in seeking to resolve contradictions.
Further, it demonstrates how backwards challengers have the situation. Namely, to resolve the contradiction, all that needs to be made is a mere suggestion of sitaution that could exist that resolves the contradiction, no matter how convoluted. No attempt even needs to be made (though attempts often are, as with the fig tree) to demonstrate that this possibility is really the case, or that it too doesn’t cause an additional contradiction.
The real problem is not specific elements of contradictions, but why the account should be accepted in the first place as, apriori, free of the normal sorts of contradictions found in texts of all sorts, especially those that are giant compilations of millenia of writing. There is, of course, no real answer to this outside of simply presuming it to be so, by faith. Seen this way, what we are left with is a text, like most others of its kind, which contains ambiguous passages that may or may not be contradictions, with no real way to demonstrate the fact for certain one way or the other. One can read them either way. The question then is WHY should we read them as necessarily all one way or the other, and indeed why should the account be believed in its entirety at all?
So, in the face of a claim of inerrancy, it will not do to simply drum up claimed contradictions (some of the most common of which are indeed, misreads, but others are clear examples of potential contradictions that can only be resolved with additional hypothetical arguments that are not present in the actual text), because that will quickly devolve. The important matter is the claim of inerrancy itself: why SHOULD anyone think that the ambiguities are not, and never can be, contradictions?