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, she 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?

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.