Matching dice bet and a double-even reroll game

(a) You and an opponent each roll two dice, four dice in all. If any two of the four dice show the same number you gain $1; if all four differ you lose $1. Is this bet worth taking?

(b) You roll one die. An even result $k$ pays $2k$ dollars and an odd result $k$ pays $k$ dollars. You may discard your first roll once and roll again, accepting the second outcome. With optimal play, what are your expected winnings?

(a) Yes — EV is $+\tfrac49$ dollar; (b) $7. (a) All four dice differ with probability $\tfrac{6\cdot5\cdot4\cdot3}{6^4}=\tfrac5{18}$, so the EV is $\tfrac{13}{18}-\tfrac5{18}$. (b) Payouts are $1,4,3,8,5,12$ with mean 5.5, so keep only 8 or 12 and reroll otherwise: $\tfrac{8+12}{6}+\tfrac46\cdot5.5=7$.