It is possible to have paper ballots that can be counted by machine for fast results, but that can also be published afterwards so that anybody can check the count, and that allow individual voters to check to see if their vote was counted correctly (but not to prove to someone else how they voted--this is called coercion-resistance), using "fill in the bubble" ballots that can be counted by widely available inexpensive optical scanner with the only new equipment needed at the polling place being special pens for marking the ballots, with all the cryptographic magic that that allows the verifiability and public counting being in the ballots themselves and put their when they are printed. For voters that aren't interested in verifying later that their own vote was correctly counted, they can ignore all that and it appears just like a regular "fill in the bubbles" process that many places have used for decades.

Here's a paper with details in PDF form [1], and here it is in HTML [2]. Here's a paper proving that it is coercion-resistant [3]. Here's a Wikipedia article [4] on it.

Voting systems with these kind of verification/auditing capabilities are called "end-to-end auditable" or "end-to-end voter verifiable" systems, and here is some information on them in general [5].

[1] https://www.usenix.org/legacy/event/evt08/tech/full_papers/c...

[2] https://www.usenix.org/legacy/event/evt08/tech/full_papers/c...

[3] https://eprint.iacr.org/2010/502.pdf

[4] https://en.wikipedia.org/wiki/Scantegrity

[5] https://en.wikipedia.org/wiki/End-to-end_auditable_voting_sy...