The dice game chuck-a-luck is played in some casinos. The rules are extremely simple. You bet $1 on a number from 1 to 6. Three dice are rolled. If your number does not appear on any dice, you lose your bet. If the number appears once, you recover your bet. If it appears twice or three times, you get $2 or $3, respectively (these amounts include your original $1 bet). What are your odds?
For bonus points, assume there are five additional friends and each person, including yourself from the original problem (so six people total), bets on a different number. Now what are the odds of winning against the house (for each better separately)?