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In a fictional screening model with 95% sensitivity and specificity, sensitivity is defined as the probability of a positive test result given the person has the disease ($P(+\mid D)$), applying to the diseased subgroup. Specificity is the probability of a negative test result given the person does not have the disease ($P(-\mid D^c)$), applying to the non-diseased subgroup. These metrics address different populations and do not directly determine positive predictive value without considering prevalence.

Conditions: The test has $95\%$ sensitivity.; The test has $95\%$ specificity.; The model distinguishes between diseased and non-diseased populations.

Gradient descent oscillates when features are not scaled because features with vastly different ranges cause the gradients with respect to their corresponding parameters to differ greatly. This leads to small step sizes for parameters of small-scale features and large step sizes for parameters of large-scale features, resulting in an inefficient, zigzagging path toward the minimum.

Conditions: Used in linear regression optimization with gradient descent.; Features have significantly different scales (e.g., one ranges from 0 to 1, another from 100 to 1000).

A set of vectors is linearly dependent if one of the vectors is redundant, meaning it provides no new directional freedom. Mathematically, this occurs when a vector lies perfectly flat on the plane (or line) already created by the others, allowing it to be written as a combination of the others.

Conditions: The set contains at least two vectors.; One vector lies in the span of the other vectors.

The main purpose is to approximate differentiable functions by ensuring that the polynomial's derivatives exactly match the function's derivatives at a specific center point. The construction guarantees that when calculating the nth derivative at $x=a$, all other terms drop out, leaving only the desired value.

Conditions: The function must be differentiable up to the desired order at the point a.; The approximation is local, centered at a specific point a.

A hexagonal prism net is composed of two hexagonal bases and six lateral faces. In the visual example, the six lateral faces appear as rectangles connecting the two hexagons in the unfolded layout.

Conditions: The solid is a hexagonal prism.; The net includes two hexagon bases and six lateral faces.

The determinant is 1. This is calculated using the formula for a 2x2 matrix where the columns are $\begin{pmatrix} 3 & 2 \\ 1 & 1 \end{pmatrix}$.

Conditions: The matrix has columns $(3,1)$ and $(2,1)$.; The determinant formula $ad-bc$ is applied correctly to the column entries.

Local linearity means that although a transformation is globally nonlinear and complex, it appears linear when viewed in a sufficiently small neighborhood around a specific point. The Jacobian matrix captures this local linear behavior.

Conditions: The transformation is nonlinear globally.; The analysis is restricted to a small neighborhood around a specific point.; The zoom level is high enough that curvature is negligible.

In Ax=b, rows of A contain the equation coefficients, x lists the unknowns, and b lists the constants. The dimensions must match.

Conditions: The system is linear.; Dimensions of the coefficient matrix, variable vector, and constant vector are compatible.

To finish on trial x, the first $x-1$ trials must contain exactly $r-1$ successes, and the x-th trial must be a success. The probability is calculated as the number of ways to arrange $r-1$ successes in $x-1$ trials, multiplied by the probability of that specific sequence.

Conditions: Trials are independent.; Success probability p is constant.; $x \ge r$.

The factors cancel because the ML inequality introduces the length of the integration path, which is $|z-a|$ for the straight line segment from a to z. This length factor multiplies the supremum term, and it exactly cancels with the $1/|z-a|$ factor that comes from the denominator of the difference quotient.

Conditions: The integration path is the straight line segment from a to z.; The difference quotient has a denominator of $z-a$, whose absolute value is $|z-a|$.; The ML inequality is applied to the integral over this segment.

$R_0=1$ is the threshold separating outbreak decline from growth. If $R_0 < 1$, each case infects fewer than one other person on average, causing transmission chains to break and the outbreak to fade.

Conditions: The video states the growth side only until immunity builds up.; The example value shown is $R_0=2.5$.

Requiring the event collection $\mathcal{F}$ to be a sigma-algebra ensures that applying the usual set operations—such as taking complements and countable unions—never produces a set outside of $\mathcal{F}$. This guarantees that if we can measure certain events, we can also measure logically related events like their negations or combinations.

Conditions: $\mathcal{F}$ is intended to be the collection of measurable sets/events in a probability space.