DID THE JEWS BELIEVE THEY COULD WIN THE FIRST JEWISH-ROMAN WAR?

C.S. Friedman, Science Fiction and Fantasy novelist
Well, we don't have psychic ability to read their thoughts, but here are some factors that played into the dynamic:
Jews did not revolt against Rome. A faction of Jews did. Not all Jews agreed with them, and on at least one occasion, some Jews sided with the Romans.
The Zealots were a group of violent extremists determined to run Rome out of Judea by making the cost of maintaining occupation too high to bear. Their best known tactic in the decades leading up to the first revolt (66 AD) were the sicarii, knife-wielding terrorists who cut the throats of Roman citizens. Yes, that's right, modern-style terrorist, striking civilians at random. (If you watch Killing Jesus, Pilate discusses a Zealot attack with Herod during the feast). Some Jews were undoubtedly behind the Zealots 100%. Others appear to have been more like, WTF? and pretty pissed that the Zealots would kill Romans and then disappear into the night, leaving more peaceful types to suffer Roman wrath. In the period between the revolts of 66 and the 130 ADs, there were various skirmishes and battles, including an all-out seige of Jerusalem. Some historians feel Judea came close to civil war, fighting with each other over whetherand how to fight the Romans.
After the Bar Kochba revolt of the 130s, we have two conflicting histories: the first from period references that suggest all the Judeans were killed, enslaved, or banished, and the other a clear historical record of a small Jewish community remaining in and around Jerusalem for centuries after the revolt, fully accepted by Rome. One might speculate that the Romans recognized the division in Jewish society, and mercilessly punished those responsible for the wars, sparing those who had not been involved in it. That is just a guess, mind you.
Now, what this means is that the driving force behind the Jewish-Roman wars in that era was a faction whose goal was not defeat of the whole Roman army, but a war of attrition, in which Judea would simply prove to be more trouble to Rome than it was worth. If the Zealots caused enough murder and mayhem and expense, eventually Rome would tire of the cost and withdraw its troops. Similar strategy has been used in various times and places in the 20th and 21st centuries, to drive out mega-powers from countries they occupied, so it is easy to envision the dynamic. The Romans were able to end their rebellion because they were a brutal and merciless empire, with no compunction about killing whole cities full of people. If they had been constrained by modern morality, who can say if the conflict would ever have ended?
That is how I read the situation, anyway. We have no witnesses from the period that speak to motivation, so we can only guess.

Hayden Siegel, In the process of making Aliyah
They did win. That is why it wasn't the 4th Servile War.
However Rome regrouped and went back in. They still did not technically win, but they for sure did not lose. They more or less got stuck in a war of attrition and Israel caved agreeing to let Herod rule over them so long as it was able to be an autonomous state of Rome. Thus it became Judeae. However Herod forsook his word and attempted to mount a Roman Eagle atop the Temple and thus the region was lost to constant rebellion. Eventually following the Bar Kochba Revolt Rome became Christianized and the Jewish people were taken as slaves. Eventually the Roman dogma started to corrupt and some day soon the world will take on a new method of governance.
The Caesar and the Senate were never great methods of government as they lent themselves to corruption just the same as their pagan icon Jesus had claimed the Temple was guilty of. The one thing the Temple and we Jews had as a method of keeping that in check was the will of G-D per-say. The High Priest did not control the people of Israel as much as say the US government does their property and there is always the chance for a Prophet to emerge to bring us back to the root or in times of the Diaspora such as now a Moshiac to led us back to our system of government.

Arturo Bonin, Amateur researcher
There were many confrontations between Jews and Romans, but two were major wars. One was the big revolt starting in 66 ce and lasting until 73 ce. This is the one you may be referring to. In retrospect, it was madness to think that the Jews would win against the mightiest empire in history, but things might have looked differently without 20/20 hindsight. Judea had a minor Roman garrison and it was quickly beaten. Probably the Jews misscalculated that the Romans, busy with internal problems and with war against the Parthians, wouldn't send a major force to reclaim Judea. Maybe they thought they'd have a repeat of the Maccabean revolt that obtained first better conditions from the Greeks and then outright independence . But that's not how rome operated. They couldn't allow a province to break free. The revolt did succeed at first and it took rome several years to reclaim it. Let's also not overlook the role that religious zealotry played in the revolt. The Jews were pushed to revolt by zealots that believed God would intervene in their favor.

Yirmiyahu Fischer, studied at Yeshiva University and continues to study Talmud
As Arturo Bonin pointed out, there were 2 major confrontations between the Jews in Judea and Rome; the revolt leading up to the Destruction of the Temple and the Bar Kochaba revolt almost 70 years later.
It should be noted that the general Rabbinic understanding is that, if the Jewish People are properly righteous, no one could stand against them, and if there are genuine spiritual problems, G-d may choose not to fully protect them.
According to the Talmud (Gitten 55b), the Rabbis opposed fighting the Romans and considered it futile. A controlling group of zealots, however, insisted that they  go and fight. Three wealthy Jews brought enough food into Jerusalem to last for a 20-year siege, but the zealots burned the warehouses in order to force the population to go out and fight the Romans. The Rabbinic leader of the time, Rabban Yochanan Ben Zakai, who was also the brother-in-law of the Chief Zealot, arranged to secretly exit Jerusalem and surrender to the Romans. In exchange, the Rabbinic Sages would be able to continue studying Torah after the Temple was destroyed. Rabbi Akiva, who became prominent many years later, criticised Rabban Yochanana Ben Zakai for not insisting that the Temple be left intact as part of the surrender terms.
In the next confrontation, Rabbi Akiva supported Bar Kochba's revolt and initially considered Bar Kochba to be the Messiah. Other Rabbis of his generation, however, disagreed.

Manuel Cantella
Of course they did. They thought that God was on their side and that the Messiah was coming to free them from the Romans. They were so sure of the victory that at the end of the war around 9000 Jews took thejr own life in despair…the Roman empire sent the elites of the army to crash the revolt and Titus with the other generals were so upset that let their troops mistreat the rebels(usually the Romans as smart conquerors demonstrated a flexible attitude toward the subjugated populations.