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Probability fixes the parameter and varies the data, resulting in a normalized distribution where the probabilities sum to 1. Likelihood fixes the data and varies the parameter, resulting in a non-normalized curve whose integral does not equal 1.

Conditions: Both views are based on the same conditional expression $P(x \mid \theta)$.; Probability view: parameter is fixed, data varies.; Likelihood view: data is fixed, parameter varies.

Probability fixes the parameter and varies the data, resulting in a normalized distribution where the probabilities sum to 1. Likelihood fixes the data and varies the parameter, resulting in a non-normalized curve whose integral does not equal 1.

Conditions: Both views are based on the same conditional expression $P(x \mid \theta)$.; Probability view: parameter is fixed, data varies.; Likelihood view: data is fixed, parameter varies.

The likelihood function is $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$. It represents the probability of observing 7 heads and 3 tails as a function of the unknown parameter $p$, the probability of heads.

Conditions: The number of tosses $n=10$.; The observed number of heads $k=7$.; $p$ is the probability of heads on a single toss.

The likelihood function is $L(p) = \binom{10}{7} \cdot p^7 \cdot (1-p)^3$. It represents the probability of observing 7 heads and 3 tails as a function of the unknown parameter $p$, the probability of heads.

Conditions: The number of tosses $n=10$.; The observed number of heads $k=7$.; $p$ is the probability of heads on a single toss.

The probability distribution sums to 1 because it accounts for all possible data outcomes under a fixed model, satisfying the axioms of probability. The likelihood curve does not integrate to 1 because it measures relative support for different parameter values given fixed data, and is not constrained to be a probability distribution over the parameter space.

Conditions: Probability view: parameter fixed, data varies over the entire sample space.; Likelihood view: data fixed, parameter varies over its domain.

The probability distribution sums to 1 because it accounts for all possible data outcomes under a fixed model, satisfying the axioms of probability. The likelihood curve does not integrate to 1 because it measures relative support for different parameter values given fixed data, and is not constrained to be a probability distribution over the parameter space.

Conditions: Probability view: parameter fixed, data varies over the entire sample space.; Likelihood view: data fixed, parameter varies over its domain.