Those are two phrases: 'better than chance' AND 'by odds of three million to one', the latter phrase modifying the first:

*... volunteers could guess what cards would be drawn randomly from a deck with a success rate that was better than chance.*

It is quite imprecise to say that the success rate was three million times better than chance, because we don't know the number of draws.

The success rate of chance for a single draw is, in this case, 1:5 (there were five kinds of cards).

For two draws, it is 1/5 x 1/5 = 1:25.

For ten draws, it is 1:9,675,625

For twenty draws, it is 1:95,367,431,640,625

If we multiply those latter ratios by three million, we get the "actual" success rates: approximately 1:3 and 1:32,000,000, respectively, which are quite dissimilar ratios.

(What I'm trying to show is that if we don't know the number of draws, we can't tell how impressive three million times better is.)