A quick survey of the literature seems to indicate that gambling activity (including lottery play) is positively correlated with income.
If it was a tax on the poor and undereducated, the correlation should be negative, yet the first three articles I found on the matter said it's positive, and that income is the strongest determinant of gambling activity.
It's more complicated than that. First of all you have to separate the different gambling activities. Some activities, like card games at a casino, are more positively correlated to income and education, while most studies on activities like the lottery and scratch tickets show negative correlation to both income and education.
Secondly simply looking at the number of times person partakes in an activity or just how much they spend only gives a small and incomplete picture. The really important number is what proportion of their total income is spent on gambling. Most studies seem to find that the poor spend a higher proportion of their income on lottery tickets than richer people.
If it was a tax on the poor and undereducated, the correlation should be negative, yet the first three articles I found on the matter said it's positive, and that income is the strongest determinant of gambling activity.