Population growth is generally exponential, not geometric, but the rate also is not consistent due to various factors.
Because Noah is a fictional character adapted from the Gilgamesh epic, the chance of him and his family populating the planet is exactly zero.
According to the fictional chronology in the Bible, the ‘flood’ occurred about 4535 years ago. This varies from JW chronology mainly due to the erroneous 20 year gap in JW chronology and the incorrect interpretation of Israel (which didn’t exist yet) ‘going into slavery’ because of that one time Isaac was teased by his brother.