Danielle may choose one of two options for the method in which she may be awarded a money prize.

option a: spin a spinner twice. the spinner is divided into four equally-sized sectors numbered 1, 4, 4, and 5. if the sum of the two spins is greater than 6, danielle is awarded $8. otherwise, shee must pay $2.

option b: flip a coin three times. if heads appears once, danielle is awarded $6. otherwise, she must pay $1.

danielle chooses the option with the greater mathematical expectation. how much more money can danielle expect to make by choosing this option over the other option?