No, you won't. See Table 1 in the paper, for 3 coin flips. You get exactly 50%, as expected because the probabilities are independent. But when you only look

No, you won't. See Table 1 in the paper, for 3 coin flips. You get exactly 50%, as expected because the probabilities are independent. But when you only look at streaks, you find that the 1s are clustered in the streaks! Thus, if you average across many sequences (aka games) and weight per-series instead of per-1, you underweight the successes that occur in sequences with longer streaks and more 1s, giving an average-of-averages that is less than 0.5. Why the researchers insist on averaging across sequences, I don't know.