The mathematical expectation is a weighted sum:
i.e. we multiply each outcome with its probability, and sum all these terms.
There are 16 possible outcomes for the spin, and here's table with wins/losses:
So, there are 9 winning spins and 7 losing spins. Since all the spins have the same probability, the probablity of winning $8 is 9/16, and the probability of losing $2 is 7/16. This leads to a mathematical expectation of
In the case of the three coin flips, all triplets have the same probability of 1/8, and the eight triplets are
TTT, TTH, THT, HTT, THH, HTH, HHT, TTT
So, Danielle wins with 3 triplets, and loses with 5 triplets. The mathematical expectation is
So, the first method is better, and the difference is 29/8-13/8 = 2.