Probabilities in the game of Quasar.

def getPrize(num):
    if num == 15:
        return 5
    elif num == 16:
        return 10
    elif num == 17:
        return 20
    elif num == 18:
        return 25
    elif num == 19:
        return 30
    elif num == 20:
        return 40
    else:
        return 0
   
def getRoll(before):
    if before > 20:
        return ((), 0)

    best_roll = ()
    best_avg = getPrize(before)

    for roll in (range(1,9), range(4,8)):
        avg = 0.
        for i in roll:
            avg += getRoll(before + i)[1]
        avg /= float(len(roll))
        if avg > best_avg:
            best_avg = avg
            best_roll = roll
    return (best_roll, best_avg)

for before in range(20):
    roll=getRoll(before)
    print("%d: %s(%f)" % (before, str(roll[0]), roll[1]))

Which leaves us with the following table of the best bets:

current sum: best choice option (average money won)
0: 4..7(24.459561)
1: 4..7(24.461427)
2: 4..7(23.844185)
3: 1..8(23.573145)
4: 1..8(23.723004)
5: 1..8(24.281559)
6: 4..7(24.718933)
7: 4..7(25.114746)
8: 4..7(23.730469)
9: 1..8(21.812592)
10: 1..8(22.166748)
11: 1..8(23.037109)
12: 1..8(24.921875)
13: 4..7(28.750000)
14: 4..7(23.750000)
15: 4..7(17.500000)
16: 1..8(14.375000)
17+: stop

Too bad the wages are too low to make it a viable source of income.