Betting on six is not justified for three different reasons.
1) Macro correctly identified the fallacy in the offset argument -- betting that since six is under-represented, then six must OFFSET (be over-represented) to achieve the proper balance per the law of averages. Macro showed that OFFSET is not necessary and that DILUTION -- given enough tries-- will accomplish the same thing.
2) Chernick correctly identified the weakness in assuming the underlying probabilities are unchanged. Chernick noted that this is a "crazy scenario". To see how crazy this, calculate the chance of getting 100 6s (or fewer) in the next 50,100 throws of a fair die. Using Excel:
- BINOM.DIST(8350,50100,1/6,1)= 5.03 E-01. Chance of 8350 or fewer (1/6th of tries).
- BINOM.DIST(8000,50100,1/6,1)= 1.26 E-05. Chance of 8000 or fewer
- BINOM.DIST(6000,50100,1/6,1)= 4.52 E-190. Chance of 6000 or fewer
- BINOM.DIST(5500,50100,1/6,1)= 2.50 E-284. Chance of 5500 or fewer
- BINOM.DIST(5400,50100,1/6,1)= 8.72 E-306. Chance of 5400 or fewer
If you had observed only 5,400 sixs, this (or fewer) would be EXPECTED if you rolled 8.72 x 10^306 sets of 50,100 die. [If the probability of an outcome is p, then that outcome is expected in n = 1/p trials.] 8.72x 10^306 is an INCREDIBLY large number of trials. Way more than billions or trillions.
Since getting 100 or fewer 6s in 50,100 tries is so unlikely, we can/should reject the null hypothesis that the underlying probability of a 6 is 1 in six, conclude that the new probability of a 6 is 100 in 50,100 and bet on anything but six [if you believe "the future is best represented by the recent past"].
3) My choice would be to not bet -- to walk away -- given the magnitude of the change and the lack of any evidence that the most recent relative-frequencies are constant.