In a six-sided dice game, a prize is won if the arithmetic mean of number rolled is 3.25-3.75. in a second game, a prize is won if the arithmetic mean is more than 4.5. in which game would you rather roll the dice 20 times or 200 times

The game with the lower mean of 3.25 to 3.75.

In either game, the expected mean of the dice would be 3.5. This can be found by adding all of the possible values and dividing that by 6.
1 + 2 + 3 + 4 + 5 + 6 = 21 / 6 = 3.5

Therefore, you are more likely to roll the lower mean than the higher mean, so you should pick that one.