DonTheDoc Pearson's explanation is wrong but answer is right.
JeydinNewWon the probability of getting one 5 only is not 5/36 but 10/36 because you can set either die to be the one with the 5 - they're separate entities, although it can be difficult to tell visibly, especially if they're identical. Instead, you could try pretending that Dice A is blue and Dice B is red or something. If we use Dice A-Dice B notation:
Setting Dice A as the 5: 5-1, 5-2, 5-3, 5-4, 5-6
Setting Dice B as the 5: 1-5, 2-5, 3-5, 4-5, 6-5
= 10/36 permutations
You are correct in saying that the chance of getting two 5s is only 1/36 - both dice must land on a 5 to satisfy the requirements for this outcome.
In summary:
Play 36 games (-$36)
10/36 plays yield one 5 i.e. $2 (+$20)
1/36 plays yield two 5s i.e. $10 (+$10)
= $6 loss
Pearson accidentally swapped the probabilities of getting one 5 and two 5s in their explanation, which only makes the question more confusing.