Gagie achieves this new bound by considering a related trick. Instead of pre-arranging the cards, you shuffle the pack and then ask your friend to draw seven cards. He or she then lists the cards’ colours, replaces them in the pack and cuts the deck. You then examine the deck and say which cards were drawn.
This time you’re relying on probability to get the right answer.
“It is not hard to show that the probability of two septuples of cards having the same colours in the same order is at most 1/128,” say Gagie.
He goes on to consider the probability of correctly predicting the sequence of cards pulled at random from a deck of a certain size and after a few extra steps, finds a lower bound on the probability of doing this correctly.