The probability of throwing a six and winning $10 is ⅙ ; the probability of a one and winning $5 is also ⅙; the probability of a four and winning $1 is also ⅙. The probability of any other number and winning $0 is ½.
Imagine the outcome of 36 games. On average there would be 6 sixes (worth $60), 6 ones (worth $30), 6 fours (worth $6). So in 36 games the winnings would be $96. If we average this out over the 36 games we get 96/36=8/3 dollars per game on average. In other words, for n games the winnings would be 8n/3 dollars. In practice, the larger n is, the closer $8n/3 would be the actual winnings. In reality there should be penalties for the non-winning numbers to balance the risk, otherwise players cannot lose.