This is really the most fundamental and important question on this site. But in my opinion, the other answers here have not done this justice by any stretch. They make it seem simple, and they may make us feel good, that we're really right, but in reality it's not so easy to show why Judaism is true, let alone prove it with certainty to "be sure" Judaism is true. They have presented more or less one-sided arguments and books, and while that is part of the puzzle, we'd be fooling ourselves to pretend that that is sufficient for determining the truth of any controversial question. One-sided arguments can just as well be presented on the Islam or Christianity SE sites, and while such an approach might satisfy some people, I don't think you'd be inclined to say that it gets them any closer to the truth.
Any answer going to the heart of determining a religion's truth is going to need to come from a neutral perspective. Before any evidence is considered, that really is the only position that makes sense: There are thousands of religions, each with firm believers. So nobody, Jews included, should start with a biased impression that they lucked out and were born into the right faith. The basic answer is going to need to explain how much evidence in favor of Judaism would be sufficient to justify belief in the faith and then examine what evidence there is. With that, this answer will attempt to approach the question from a rationalist perspective.
How much evidence is needed
For many things in life, not that much evidence is needed to believe something. You can tentatively believe that your generally honest friend had waffles for breakfast going on their casual word alone, because there's nothing extraordinary about his claim. If a stranger tells you he just ate a giraffe sandwich though, you'd probably need an explanation for how he obtained such rare meat along with some good reason to believe him like photographic evidence. If an Ancient Roman tells you that he saw Romulus ascend to heaven and he has a whole crowd of witnesses to prove it, you'd need to do some thorough cross examination of that crowd, and even if it checks out many people would remain skeptical. For Judaism, it is more equivalent to the latter cases. Although for some of us who were raised in a Jewish environment it may not seem like a particularly extraordinary claim, from the perspective of a person that doesn't already believe, they may need this higher level of evidence. For an outside observer, it can come off as an extraordinary claim that a given religion is true and that stories of supernatural events did happen and that the Jewish people had the unique ability to properly maintain stories without modification over a hundred generations, particularly since it is controversial in the sense that secular people have their own arguments against it. So a reasonable response to those arguments along with substantial evidence or reasoning is what is needed to be sufficiently confident that Judaism is true.
I'll bring up some of the more significant challenges to Judaism and how they might be resolved, and I will discuss the more popular arguments in favor of Judaism and where they might fall short. And so while this answer won't definitively prove Judaism true, I hope it will be useful to help people make that determination for themselves.
Arguments against Judaism and responses
Natural and early history according to the Torah appears wrong
One of the tougher challenges to Judaism I've seen basically argues that the Torah errs in describing early history, especially in the first 11 chapters of Genesis describing creation, the flood, and the development of language. For example, the scientific evidence indicates that the universe and Earth are older than 6000 years, that Noah's flood never occured, and that human history predated Adam and was not interrupted by Noah's flood. It is not a strong position, and possibly theologically unsound to dismiss the meaning, especially as it is understood as literal in various places in the Talmud. Further, the Torah records specific genealogies dependent on the historicity which are not well suited for allegorizing. Some Rishonim such as the Meiri also prohibit interpreting the story of creation as allegory, as discussed in this Machzikei HaDas essay on the varying opinions regarding allegorical interpretation of scriptures. Non-literal meanings are also made tenuous by the writings of other Rishonim, such as Rambam who says that it is a fundamental principle of the Torah that Adam was the first human created and that the genealogies and language development described in Genesis are correct in Guide for the Perplexed, Part III Chapter L and Rabbi Yehuda Halevi who says that evidence of pre-Adamic human civilization would weaken his belief in Kitab al Khazari, Part One paragraphs 60-61.
Among the suggestions I’ve seen to address this are that perhaps the scientists are severely mistaken, perhaps miraculous events would include aspects that obfuscate evidence of their occurrence, or that some opinions such as Rav Saadia Gaon do indeed allow for non-literal interpretations of these narratives (although only as a last resort, again as discussed in the Machzikei HaDas essay), either in total (e.g. "It teaches a deeper lesson") or in part (e.g. "Adam was the first man in a spiritual sense" or "Noah's flood was regional").
The Torah and Tanach appear to have contradictions
Another challenge is that the Torah and Prophets (though not necessarily the Writings) are supposed to be inerrant, yet some stories and verses appear to contradict with one another. For example, Numbers 33 and Deuteronomy 10 report conflicting versions of the journey through the wilderness and Aaron's death. In I Samuel 15 all of Amalek is killed, but by I Samuel 30 they’re pillaging Jewish towns. According to the genealogy in Genesis 46 Benjamin had 10 sons, according to Numbers 26 he only had 5, and there are differences about this as well in I Chronicles 7 and 8. II Kings 25 and Jeremiah 52 have various differences, such as whether Evil-merodach elevated and freed King Jehoiachin on the 27th of the month or on the 25th of the month. There are more examples, especially contrasting Chronicles (though being in Writings it may not have to be inerrant) with the narratives in the Prophets. So if the verses contradict, the Tanach must not be a reliable text, and it must not be that it was truly divine and inerrant.
Of course, Talmudic sages and later commentaries do offer resolutions to these contradictions, for example by saying that in one of the verses it meant something different from how it appears. Aaron's burial location in Deuteronomy was figurative as a rebuke to the Jews. Perhaps some of Benjamin's families had been totally wiped out before the Numbers census and genealogy. And so on. Since there are ways to say that these verses don't actually contradict, they therefore may not be a product of human authorship and error. Rather, such apparent contradictions may be intentional, explained in the Oral Law, possibly to convey deeper meanings.
The Torah may include anachronisms
This challenge is basically to say that the Torah makes certain references to places or things or events that shouldn't be in there and that would make more sense if it had been compiled over the roughly 900-400 BCE time frame that Bible critics propose. For example, in Genesis 11:28 and other places, Abraham is said to be from Ur Kasdim, which means Ur of the Chaldeans, yet it was not called the land of the Chaldeans until the 9th century BCE, and they could not have been around at all until at least the 11th century BCE. Or in Exodus 1:11 the Jews are credited with building Pithom and Ramses. Yet these cities were built later, for example Ramses was built for Ramses II who didn't reign till 1279 BCE, well after the Jews had left Egypt (which was 1313 BCE according to Seder Olam or even earlier factoring in the missing years).
There are various other examples in the Torah and Prophets. But again there are responses. Maybe the archeologists are mistaken, or maybe references in the Torah should be understood differently from how one might initially think. For example, maybe the reference to Ramses in the Torah was to a place not known to archeologists, and Egyptian rulers happened to have that name and a similarly named city afterwards. So while these details may be challenges to address, they don't necessarily make an earlier, legitimate authorship of the Torah untenable.
The Talmud makes erroneous statements
Another challenge to Judaism is to say that the sages in the Talmud made mistakes, and at least by some views, it is necessary to say that the sages couldn't have made mistakes, and even if they could, it doesn't speak well for their reliability in general. From the issue of the missing years where the Talmud considers the Second Temple period to have been significantly shorter than what is known from archeology of the time, to some views expressing a flat-earth cosmology, to apparently indicating that lice spontaneously generate, errors show the sages to be flawed. But these errors can also be explained within Judaism. Rabbi Natan Slifkin is noted for his addressing of discrepancies between chazal and science. By some opinions, you are required to believe that the sages did not make mistakes, in which case you may consider Talmudic errors to simply be allusions to their actual, deeper meanings. By other opinions, the sages are not considered infallible, and the only science that it matters for them to get right is when it relates to Halacha.
These are some of the big ones, but I have seen people presenting additional challenges, arguing that prophesies in the Tanach haven't come true, that textual oddities suggest human authorship, that stories and practices in the Torah are derivative of other ancient religions, and so on. Discussing and answering them all would be outside the scope of this answer. In general, the degree to which these and any other challenges to Judaism seem significant, and the degree to which their resolutions are satisfactory, will come down to a matter of opinion. It is an individual's responsibility to look into the facts and make a judgment of how innocuous or problematic such challenges may be and what degree of evidence in favor of Judaism is needed to outweigh them.
Arguments for Judaism
There are a wide array of arguments in favor of Judaism. Kiruv resources have gathered a variety of pieces of evidence and logical proofs. Some of the arguments are better and more common than others, while still more are unfortunately based on misinformation and don't serve Judaism well.
I just think it's worth touching on this: A lot of proofs claim to show an amazing piece of knowledge contained in the Torah or Talmud or describe an amazing paranormal event but aren't really backed up by the facts. Unfortunately, they often get passed around without scrutiny, and that is counterproductive. I'm not saying that all amazing claims are necessarily false. But before something like that is repeated, it should be fact checked first to rule out misinformation.
Personal Experience and Miracle Stories
A fairly common argument that individuals use as evidence in favor of Judaism is stories of miracles or Hasgacha Pratis (divine providence). This is basically the category of supernatural coincidences. Often times they are a personal and anecdotal way of demonstrating evidence of Judaism to one's self or close peers. On the other hand, due to the nature of these stories, examples that impress oneself or a person's peers may not convince society as a whole. Some examples of stories I've heard involve prophetic dreams, precognition, miraculous healing, divine providence, and so on, and they go to attribute these events to God and take as evidence for Judaism.
The limitations of this line of evidence, though, are (1) that without having a personal experience yourself, you may not trust the supernatural event truly happened in the way the storyteller suggests, (2) that similar miracle stories are also commonly used by Christians, Hindus, and people of other religions, so unless Jewish stories can be meaningfully and qualitatively distinguished from other ones, it either is evidence of God but not a particular religion or it may not be a reliable way of determining the truth of something, and (3) these things may often be explainable by statistical expectations and cognitive shortfalls.
Another line of arguments I've seen are on Bible Codes. Basically, when taking all the letters (without spaces) in the Torah (and Tanach too), starting with one letter and skipping a fixed (and ideally small) number of letters can produce new words, or skip codes. The idea is that there are groups of skip codes in sometimes relevant passages for related ideas that describe people or events. The idea is that these hidden codes show that the exact text of the Torah and Tanach were guided by an omniscient force. Although skip codes have no reliable predictive applications, proponents argue that there are statistically significant skip codes describing people or events which is enough to suggest divine authorship, which in turn implies that Judaism is true.
Not everyone is convinced by this, disputing the statistical significance, saying that skip codes can be found in any sufficiently large set of letters and that a skip code for anything can be found in the Torah (just as with other large works), including events that did not happen, and lastly that our modern Torahs may not have the original spellings which would in many cases erase the traces of intentional skip codes if there ever were any. It may indeed be that there are statistically significant hidden codes in Tanach demonstrating divine authorship, but it's hard to verify if that's really the case.
Archeological evidence is another point that people use as evidence that the Torah narrative is true. Permission to Receive by Lawrence Kelemen has a chapter called the Empirical Issue which basically is a collection of archeological finds of ancient places or names or events that seem to corroborate the Torah's narratives. For example, an Ancient Egyptian story called The Tale of Two Brothers has similarities to the narrative in the Torah of Joseph and Potiphar's wife, and the Ipuwer Papyrus from Ancient Egypt seems to describe some of the 10 plagues of the Exodus from Egypt.
Such archeological claims are often disputed, however. For example, most scholars date the Ipuwer Papyrus to several centuries before the Exodus from Egypt would have been, and it seems to describe events that don't really match the 10 plagues. Using archeology can also be a double-edged sword, as mainstream archeologists tend to consider findings in Ancient Israel to be largely contrary to the slavery, exodus, conquest, and sometimes even the unified kingdom narratives. Without personal expertise, it may be hard for an individual to judge which archeological perspectives are more accurate, so this may also not be the easiest avenue to argue for Judaism.
National Tradition (aka the Kuzari Argument)
The "Kuzari Argument" may be the most popular argument in favor of Judaism. It is also a cornerstone of the Permission to Receive book described above. Succinctly, the Torah describes miraculous events experienced by the whole Jewish nation, and each generation learned about this from the previous. No generation would have accepted the Torah if it had just been made up by someone, especially since it contains a lot of doctrines and rules, because they could check with their elders to verify whether the stories were true. Therefore the Exodus narrative, and by extension the Torah received in it, are true.
Even this though is not a perfect proof. It depends on some assumptions, that the ancient Israelites were sufficiently incredulous, that at no point did a ruler force beliefs onto the people, that mythologies can't evolve much, that any religion the people might have had prior to the Torah would have been easier, and that at no point could a false, allegedly forgotten, history be "reintroduced" to the people. These assumptions are open to contention, and some of these assumptions may even conflict with narratives within the Tanach itself.
For example, it may be such mythologies really could develop naturally. Some argue that other cultures such as the Aztec and Sioux have national miracle myths. The narrative in Tanach may also not necessarily involve a continuous national tradition, as there are various places that state that at times God was forgotten and monotheism was rejected, for example:
And also all that generation were gathered unto their fathers; and there arose another generation after them, that knew not the LORD, nor yet the work which He had wrought for Israel. (Judges 2:10)
Further, perhaps a ruler could have indeed instituted religious reforms involving revisionist history, by authoritarian force and/or by claiming the history was forgotten. For example, King Asa is described as implementing with force religious reforms to spread monotheism and Torah to the people:
And they entered into the covenant to seek the LORD, the God of their fathers, with all their heart and with all their soul; and that whosoever would not seek the LORD, the God of Israel, should be put to death, whether small or great, whether man or woman. (II Chronicles 15:12-13)
Perhaps more relevant would be II Kings chs. 21-23, where a long period of time is described where the Jewish people were alienated from the Torah and practiced polytheism until it was discovered again by King Josiah’s scribe and High Priest. As a result, Josiah instituted religious reforms and restored monotheism, known as the Deuteronomic Reform, which may have been an opportunity for the king to introduce the people to a revised history of a national revelation under the pretense that it had been forgotten under the previous generations. Later the Tanach indicates that his reforms did not fully take hold until generations later when Ezra's followers finally seemed to have monotheism locked in. However, even his followers were largely ignorant of the contents of the Torah, as described in Ezra and Nehemiah, and some argue that this may have been another possible opportunity for new or altered narratives to be added to scriptures before canonization.
Despite such challenges, defenders maintain that the national miracle traditions really can't be faked and those Aztec and other counterexamples are not comparable. Some also argue that despite select verses which suggest a gap in tradition, other narrative details suggest that there must have always been at least some faithful tradition keepers. In the end, some consider the argument persuasive, while others do not.
When it comes down to it, some people look at the evidence and are persuaded that Judaism is indeed true, others may have the complete opposite takeaway, and still more may have middling confidence. For this last group, an argument that they use to justify practice of Judaism is Pascal's Wager (which is an idea I heard expressed by Rabbi Dovid Gottlieb in conjunction with his version of the Kuzari argument). Their reasoning goes that they're not sure if the religion is true or false and that it's better to practice Judaism on the chance that it will help them in the afterlife. How good of a reason this is to practice Judaism may depend on how likely they think it is that Judaism actually is true, whether there are practical downsides to Judaism, what the consequences would be for a soul that doesn't practice Judaism, and (if the person thinks an alternate religion might also be true) what the consequences might be for following a false religion. So Pascal's Wager can often serve as a reason to follow Judaism. But it won't help anyone conclude whether Judaism is actually true.
There are, of course, various other reasons or pieces of evidence that are used to argue in favor of Judaism. Some argue for it on account of the survival of the Jewish people, the establishment and military successes of the State of Israel, numerology, the fact that the First and Second Temples were destroyed on the same calendar day, and so on. But again it is outside the scope of this answer to discuss them all, and it is up to the individual to explore and analyze their strengths and weaknesses.
Are these arguments sufficient to justify belief in Judaism? You will have to judge that for yourself. To that end, the statistical equation known as Bayes’ theorem can be especially useful in calculating the likelihood that Judaism is true by mathematically considering the various pieces of evidence. (See the addendum below for more information on using Bayes’ theorem.)
Based on the arguments I've seen, a case for Judaism can be made, yet it is by no means easy to definitively prove that Judaism is true. As far as why that should be, assuming of course Judaism is indeed true, other questions try to explore that as well as why God would require us to be religious when there are these doubts.
But as far as knowing "for sure" if Judaism is true, the fact is that there are arguments in favor of it but arguments against it. And what convinces one person might not convince another. It is up to every individual to examine the evidence and do their best to answer the question as honestly as they can.
Using Bayes' theorem to calculate the probability of Judaism
Statistics can often be counterintuitive, so Bayes' theorem is helpful for factoring in pieces of evidence to estimate the probability that something is true. Specifically, it works by multiplying the prior odds that a hypothesis is true by the probability of there being a given observation if the hypothesis were true, divided by the total probability of there being that observation. There are various online resources explaining it more, but one helpful video is A Visual Guide to Bayesian Thinking by the rationality public speaker Julia Galef.
A caveat here is that for topics such as this, it relies largely on an individual's best assumptions rather than clearly known probabilities. And due to differences in the knowledge different people have about various issues, or personal biases that skew what they consider reasonable, what one person considers to be a reasonable probability to input may differ from that of another. Consequently, although this can be useful, it will still not produce a single, definitive value for all people. Individuals must also be mindful of potential biases (e.g. if they think an expected probability for a piece of evidence is extremely dependent on Judaism being true, they should ensure they know enough about the issue to honestly justify that confidence). Of course, this is still far more reliable and useful than a simple gut feeling, which is all anyone can reasonably ask for.
The equation can be written like:
P(J|E) = P(E|J) x P(J) / [P(E|J) x P(J) + P(E|~J) x P(~J)]
P(J|E) is the posterior probability that Judaism is true,
P(E|J) is the probability of specific evidence existing assuming Judaism is true,
P(J) is the prior probability that Judaism is true before considering that evidence,
P(E|~J) is the probability of the evidence existing assuming Judaism is not true, and
P(~J) is the probability that Judaism is not true (which equals
1 - P(J)). The value for a probability can range from 0 to 1 (e.g. 1 = 100%, 0.05 = 5%, etc.).
The equation in this form only factors in a single argument (or piece of evidence, or observation), so to get a final estimate, you would need to take the result of the equation, treat that as the new prior probability, and repeat the calculation for the next argument. (The order for calculating the various arguments does not matter for conditionally independent observations like this.) You don't have to factor in every argument there is, just the significant ones, both for and against, where whether Judaism is true or false would make a significant difference in your expectation of those arguments existing.
Example using Bayes' theorem
I'll give an example of using the equation, but I cannot run the actual calculations on behalf of others as the numbers I would offer would be based on my personal knowledge and estimates and colored by my personal biases. So for illustrative purposes only I will give an example using values that I do not necessarily endorse:
So first, we need to come up with an initial
P(J). That is, if you didn't have any evidence for or against Judaism, what are the odds that it is the one true religion? There are thousands of religions today, about twenty that have millions of followers. A person might consider all religions to be equally plausible, or a person might know certain things about some of those religions to discount them. Some may think it doubtful that there even is a true religion. Just for our example let's say a person thinks there is a 50% chance that there is a true religion among the largest 20 religions, and they do not initially necessarily consider any one of those religions to be more likely than any other, so we'll set the prior probability that Judaism is true as
0.5 * 1 / 20, or
P(J) = 0.025 and
P(~J) = 0.975.
For an argument to factor in, let's do the establishment of the predominantly Jewish state and government of Israel. Let's say that you would have only expected a 5% chance that the Jewish people could have succeeded in establishing a state naturally, whereas if Judaism were true you would expect maybe a 50% chance of God enabling the establishment of a state in this way. So we say
P(E|J) = 0.5 and
P(E|~J) = 0.05 leading us to:
P(J|E) = 0.5 * 0.025 / [0.5 * 0.025 + 0.05 * 0.975)]
P(J|E) = 0.204
Considering a first piece of evidence in this way raised our probability that Judaism is true from 2.5% to 20.4%, which is a significant change. Next let's say we want to factor in the effect of the Kuzari argument. Let's say we find the argument fairly compelling, expected if Judaism were true and fairly unlikely that it would have come about otherwise, and we may judge that
P(E|J) = 0.9 and
P(E|~J) = 0.1. And now we can also set a new prior probability from our posterior probability result of the last equation, so
P(J) = 0.204 and
P(~J) = 0.796. Which means:
P(J|E) = 0.9 * 0.204 / [0.9 * 0.204 + 0.1 * 0.796]
P(J|E) = 0.698
We now have gone up to a probability that Judaism is true of 69.8%. To wrap up our example we'll factor in a counterargument, apparent anachronisms in the Torah. Let's say that a later authorship may lead to some anachronisms, but also that it's plausible that those references in the Torah have been misunderstood or the archeologists would make mistakes. We might decide
P(E|J) = 0.5 and
P(E|~J) = 0.75. And from the previous calculation,
P(J) = 0.698 and
P(~J) = 0.302. So:
P(J|E) = 0.5 * 0.698 / [0.5 * 0.698 + 0.75 * 0.302]
P(J|E) = 0.606
With this, the posterior probability that Judaism is true has decreased slightly to 60.6%. And to continue refining our value of the probability that Judaism was true, we would continue doing this math until all the significant evidence is considered and we have a useful final calculation that represents how likely we believe it is that Judaism is true.
This should only be considered as an illustrative demonstration of how to utilize Bayes' theorem to figure out how likely you believe it is that Judaism is true. But do the math as modeled above, using values that you yourself consider reasonable, and it will help you reach a useful number to tell you how likely it is Judaism is true.