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The first three running averages are 55, 60, and 55. The first average is just the first observation (55).

Conditions: Observations are 55, 65, and 45.; The sample mean is calculated cumulatively.

The first three running averages are 55, 60, and 55. The first average is just the first observation (55).

Conditions: Observations are 55, 65, and 45.; The sample mean is calculated cumulatively.

The sample mean converges to 50 because 50 is the expected value $E(X)$ of the random variable $X$ (number of heads in 100 fair-coin tosses). According to the law of large numbers, as the number of trials $n$ approaches infinity, the sample mean $\overline{X}_n$ approaches the expected value.

Conditions: The coin is fair.; $X$ counts heads after 100 tosses.; $n$ approaches infinity.

The sample mean converges to 50 because 50 is the expected value $E(X)$ of the random variable $X$ (number of heads in 100 fair-coin tosses). According to the law of large numbers, as the number of trials $n$ approaches infinity, the sample mean $\overline{X}_n$ approaches the expected value.

Conditions: The coin is fair.; $X$ counts heads after 100 tosses.; $n$ approaches infinity.