A brief outline of the evidences that convince me of the probable truth of Christianity.
To begin with, there is strong antecedent evidence (such as fine tuning) for the existence of God. And then there is substantial direct evidence for the reliability of the New Testament:
1. Every ancient source who comments on the matter agrees that the Gospels were written by their traditional authors: Matthew and John by the disciples of those names, Mark by a disciple of Peter as a record of Peter's preaching, and Luke by the companion of St. Paul. This is true up until the skeptic Faustus in the 4th century. And even Faustus reports no rival tradition of authorship. (There is also broad though not unanimous agreement on the traditional order of Matthew, Mark, Luke, John.) These traditional authors would all be in a good position to accurately report the life of Jesus, as either eyewitnesses of the events of the Gospel or people with access to eyewitnesses.
2. Counterarguments in favor of late authorship by non-eyewitnesses tend to rely on more subjective judgments about what a particular author writing at a particular time would be likely to write, or how one author would copy another source, and so on. These kinds of judgments seem to me less reliable than the testimony of ancient sources.
3. There are many specific "external" evidences of the general reliability of the Gospels-Acts: specific details about 1st-century Judea (geography, politics, etc.) that the Gospels-Acts get right that would be difficult to get right if they weren't writing real history. This is especially true for Luke-Acts.
4. There are more general correspondences of the Gospels-Acts with what we know of the historical period that would be unlikely to result except from historical accounts. For example, there is a close correspondence between the frequency of Judean names around that time in other sources and their frequency in Gospels-Acts: common names are common in both, rare names are rare in both, etc. For instance, Mary was the most popular female name at the time and it is the most common female name in Gospels-Acts. The level of correspondence here is much different from what we see in, for example, modern novels about the period like Ben Hur.
5. There are "internal" evidences of textual reliability, especially "undesigned coincidences," where two or more texts fit together in a way unlikely to result from copying but likely to result from independently reporting history. For example, all four Gospels tell us of the feeding of the five thousand, and they all tell us that Jesus turned to his disciples to ask, “Where shall we buy bread for all these people?” Only John 6, though, tells us that he addressed this question to a particular disciple, namely Philip -- a minor disciple who is hardly mentioned elsewhere in the Gospels except in the calling of the 12. Meanwhile, Luke 9 tells us that the feeding took place in the town of Bethsaida, something John doesn’t tell us. And in John 12, in a completely different context, we independently learn that Philip was from Bethsaida. If the Gospels are reporting history, these other passages explain why Jesus spoke to Philip and not one of the other disciples: it's natural that Jesus would ask someone from Bethsaida where to buy food if they were in Bethsaida. But we wouldn't expect these details to coincide if John was copying Luke, or they were independently copying a third source (unless perhaps that third source was an extremely detailed novel with the level of care and precision of the Lord of the Rings; but that kind of text didn't exist at that time).
6. When we look at the apocryphal Gospels, we find their character to be very different from the canonical Gospels on the above scores. Even at a superficial level, they rarely go into similar detail about people and places, and when they do they almost always pick famous people or places like Peter or Jerusalem.
7. The above suggests a backdrop of the Gospels-Acts being authored by highly reliable and careful historians. Notably, some of the evidences for reliability involve miracles, like the feeding of the five thousand. With this background, the testimony of the Gospels, not to mention the other books of the New Testament, in favor of the resurrection makes this particular miracle highly probable. The empty tomb, the appearances of Jesus to the apostles, the conversion of St. Paul, the willingness of the disciples to go to their deaths in attestation of the resurrection, and the rapid growth of the early Church all contribute to this case. But a pop-apologetic minimal facts arguments is liable to mislead inasmuch as it presents these facts as isolated evidences rather than against the above backdrop for the reliability of the New Testament.
Taken together, I think the evidence for the general reliability (=/= inerrancy) of the New Testament is extremely strong, including in its miracle reports. Various Christian denominations add a lot more to this in terms of doctrine, of course. But I think the above makes it highly probable at least that the ancient Christian creeds are approximately true.
This is essentially Hume's argument: that claims to a miracle are mistaken is always more probable than that a violation of an otherwise uniform law of nature actually occurred.
But there's no viable theory of probability on which this principle comes out true. For any prior probability that is not 0, it is in principle possible to get evidence that raises that probability to above .5. We can see this intuitively because Hume's principle would rule out the possibility of ever disconfirming a previously uniformly confirmed scientific theory. For a simple example, we would have to conclude that European explorers were mistaken in reporting black swans in Australia, since all previously observed swans had been white.
That doesn't mean this particular evidence is sufficient to raise the probability of the resurrection above .5. But there's no in principle impossibility to getting evidence that makes it more likely that a man rose from the dead than that that evidence is misleading in some way.