The gambler makes 100 bets and wins 10. How much money does he have at the end?
Odds of 8 to 1 means that for every \$1 you wager, you could either
- win \$8 in addition to receive your \$1 wager back
- lose your \$1 wager
So from his 10 wins, he gets \$80, and from his 90 losses, he is down \$90. Net, he is down \$10.
Odds of 8 to 1 mean you make a bet of \$1, then you win \$8. You also receive your original \$1 back, so if you win an 8 to 1 bet, you pay \$1 and receive back \$9.
In regards to your problem, the gambler pays in $\$100$, and gets back $\$90$, so all in all, he has lost $\$10$.