Bunuel wrote:

Official Solution: 20 throws of a die produces following results:

What is the probability that one more throw to this series will increase the mean score?

Both the scores 1 and 6 have occurrences of 4, they cancel out to an average of 3.5 with half the roll outcomes increasing the mean score.

To save time we can average scores 2-5, or we can reason that since scores 4 and 5 have a total of 4 occurrences, while score 2 only has 3 occurrences, that the average will be just a bit greater than 3, and that only rolling 4, 5, and 6 will increase the average.