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The bars represent the probabilities of all possible outcomes (head counts) for a fixed model. Because they exhaust the entire sample space, their sum must equal 1, which is the defining normalization property of a probability distribution.

Conditions: The coin model is fixed at $p=0.5$.; The number of flips is 10.; $k$ indexes all possible head counts from 0 to 10.

The bars represent the probabilities of all possible outcomes (head counts) for a fixed model. Because they exhaust the entire sample space, their sum must equal 1, which is the defining normalization property of a probability distribution.

Conditions: The coin model is fixed at $p=0.5$.; The number of flips is 10.; $k$ indexes all possible head counts from 0 to 10.

The expression $P(x \mid \theta)$ has two interpretations depending on which variable is held fixed. If the parameter $\theta$ is fixed and the data $x$ varies, it represents a probability over data under a known model.

Conditions: For probability: $\theta$ is fixed and $x$ varies.; For likelihood: $x$ is fixed and $\theta$ varies.

The expression $P(x \mid \theta)$ has two interpretations depending on which variable is held fixed. If the parameter $\theta$ is fixed and the data $x$ varies, it represents a probability over data under a known model.

Conditions: For probability: $\theta$ is fixed and $x$ varies.; For likelihood: $x$ is fixed and $\theta$ varies.