Problem #PRU-65353

Problem

We throw a symmetrical coin $n$ times. Suppose that heads came up $m$ times. The number $m/n$ is called the frequency of the fall of heads. The number $m/n-0.5$ is called the frequency deviation from the probability, and the number $|m/n-0.5|$ is called the absolute deviation. Note that the deviation and the absolute deviation are random variables. For example, if a coin was thrown 5 times and heads came up two times, the deviation is equal to $2/5-0.5=-0.1$, and the absolute deviation is 0.1.

The experiment consists of two parts: first the coin is thrown 10 times, and then – 100 times. In which of these cases is the mathematical expectation of the absolute deviation of the frequency of getting heads is greater than the probability?