I previously shared data from Pew’s Economic Mobility Project about social mobility (I, II). As far as income mobility goes, you are 10x more likely to wind up in the richest fifth as an adult if you were born there than if you were born in the poorest fifth.
As far as wealth mobility goes, you are more than 5x more likely to wind up in the wealthiest fifth as an adult if you were born there than if you were born in the least wealthiest fifth.
One convenient way to describe what’s going on is that rich kids are more likely to get a better education, which translates into being richer and wealthier as adults. It is certainly the case that richer kids are more likely to get a college degree, and it is certainly the case that getting a college degree leaves you much better off on average than not getting one. But this does not explain the full picture of social immobility. Take a look at this super-complicated chart, which I will describe below.
This is basically the same graph as the income mobility one above, except it breaks down income mobility both by a kid’s economic starting point and education. So the red bars all go together and show where kids born into each income quintile who do not graduate college wind up as adults. The blue bars show the same thing, but for kids who do graduate college.
Look at the red bar furthest to the right. That is the bar describing where kids born into the richest fifth who do not get a college degree wind up. Notice that 25% of those kids still wind up in the richest fifth. Now look at the blue bar furthest to the left. That is the bar describing where kids born into the poorest fifth who do get a college degree wind up. Notice that only 10% of those kids wind up in the richest fifth.
So, you are 2.5x more likely to be a rich adult if you were born rich and never bothered to go to college than if you were born poor and, against all odds, went to college and graduated. The disparity in the outcomes of rich and poor kids persists, not only when you control for college attainment, but even when you compare non-degreed rich kids to degreed poor kids!
Therefore, the answer to the question in the title is that you are better off being born rich regardless of whether you go to college than being born poor and getting a college degree.