Sod Ha'ibur
— Origin of the Hebrew Calendar
— Conflict between Rabban Gamliel and Rabbi Yehoshua

The National Aeronautics and Space Administration (NASA) provides a 6000 Year Catalog of lunar data including the conjunction time of every new moon. 

Against that backdrop we can use Hebrew Calendar Basics to show the Molad of Tishrei.

Hebrew Calendar rules mandate that if the Molad occurs on any given day then that day is considered as the start of the month. 

According to the Rambam the moon will not be seen for approximately 24 hours after the conjunction. The new moon can only be seen in a narrow window of time between 6:00 PM after the sun has set and 7:00 PM but before the moon has set. If, for example, the conjunction is at 10:00 AM then the new moon will not be seen that evening but will be seen some 32 hours later after sunset.

The table of the Molad of Tishrei is from 101 CE through 2100 CE. No Dehiyot (postponements) were incorporated in this table as they were establish after out period of interest, i.e. after 358 CE.

From the table are the following cogent facts.

#1 The Hebrew Calendar Calculation was designed so that the actual moon could never be seen on Rosh HaShanah. 

#2 70.4% of the time we will act as if the new moon was seen two days before witnesses could see it. The initial case in 2002 was indeed the norm.

#3 75 times in the entire 2000 year period there was a three day discrepancy between when the Molad designated that Tishrei should start and when witnesses would have been able to see the moon.

It will be shown that a three day discrepancy is synonymous with an Old Moon visible on the morning of Erev Rosh HaShanah in: 

120 CE   The conflict of Rabban Gamliel and Rabbi Yehoshua

198 CE   The 4 mil walk by the light of the Old Moon by Rabbi Hiyya

358 CE   The date erroneously assumed by many to be when this calendar was created by Hillel II.

835 CE   The letter of the Exilarch

Here are these entries from the table