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The solution is checked in the other original equation to confirm that the ordered pair satisfies the entire system, not just the single equation used for back-substitution. If the pair was found by substituting into the first equation, checking it in the second equation verifies that both equations are simultaneously true for those values.

Conditions: A candidate ordered pair has been found.; At least one original equation remains available for substitution checking.; The equation used for back-substitution is not reused for the check.

The solution is checked in the other original equation to confirm that the ordered pair satisfies the entire system, not just the single equation used for back-substitution. If the pair was found by substituting into the first equation, checking it in the second equation verifies that both equations are simultaneously true for those values.

Conditions: A candidate ordered pair has been found.; At least one original equation remains available for substitution checking.; The equation used for back-substitution is not reused for the check.

The solution for the state price vector is unique because the coefficient matrix of the resulting linear pricing system has a nonzero determinant. A nonzero determinant guarantees that the system of linear equations has exactly one unique solution for the unknown state prices.

Conditions: The market model has three states and three traded assets.; The pricing system is a 3x3 linear system.; The coefficient matrix of the system is nonsingular (determinant is nonzero).

The solution for the state price vector is unique because the coefficient matrix of the resulting linear pricing system has a nonzero determinant. A nonzero determinant guarantees that the system of linear equations has exactly one unique solution for the unknown state prices.

Conditions: The market model has three states and three traded assets.; The pricing system is a 3x3 linear system.; The coefficient matrix of the system is nonsingular (determinant is nonzero).

The Euclidean algorithm terminates because the remainder in each step is strictly smaller than the previous divisor. Since the second argument is a nonnegative integer that decreases with each step, it must eventually reach zero.

Conditions: The algorithm operates on nonnegative integers.; Each step replaces the pair $(a,b)$ with $(b, a\bmod b)$.; The remainder $a\bmod b$ is strictly less than $b$ when $b > 0$.

The Euclidean algorithm terminates because the remainder in each step is strictly smaller than the previous divisor. Since the second argument is a nonnegative integer that decreases with each step, it must eventually reach zero.

Conditions: The algorithm operates on nonnegative integers.; Each step replaces the pair $(a,b)$ with $(b, a\bmod b)$.; The remainder $a\bmod b$ is strictly less than $b$ when $b > 0$.

Because adding the modulus 12 to -2 gives 10 ($-2 + 12 = 10$). This means -2 and 10 represent the same residue class modulo 12.

Conditions: Work modulo 12.; Allow negative representatives.

Because adding the modulus 12 to -2 gives 10 ($-2 + 12 = 10$). This means -2 and 10 represent the same residue class modulo 12.

Conditions: Work modulo 12.; Allow negative representatives.

In Example 1, there is a lot of variation within each group but not much variation between the groups. Because most of the difference is due to the people rather than the drink, the video concludes that the drink did not make much of a difference, leading to accepting the null hypothesis.

Conditions: The displayed Example 1 data are used.; Within-group variation is much larger than between-group variation.

In Example 1, there is a lot of variation within each group but not much variation between the groups. Because most of the difference is due to the people rather than the drink, the video concludes that the drink did not make much of a difference, leading to accepting the null hypothesis.

Conditions: The displayed Example 1 data are used.; Within-group variation is much larger than between-group variation.

Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.

Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.

Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.

Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.

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 decision rule is based on comparing the class average to a fixed threshold. An outlier artificially inflates the average above the threshold, preventing the class from receiving help.

Conditions: The decision rule is: hold extra classes only if the average score is below 35.; The dataset contains an outlier.; The threshold is fixed at 35.

The decision rule is based on comparing the class average to a fixed threshold. An outlier artificially inflates the average above the threshold, preventing the class from receiving help.

Conditions: The decision rule is: hold extra classes only if the average score is below 35.; The dataset contains an outlier.; The threshold is fixed at 35.

The expected number of heads is 50 because the expected value for a counting experiment like this is calculated as the number of trials multiplied by the probability of success on each trial. With 100 tosses and a fair coin (probability of heads = 0.5), the calculation is $100 \cdot 0.5 = 50$.

Conditions: The coin is fair.; There are 100 tosses per trial.; $X$ counts heads.

The expected number of heads is 50 because the expected value for a counting experiment like this is calculated as the number of trials multiplied by the probability of success on each trial. With 100 tosses and a fair coin (probability of heads = 0.5), the calculation is $100 \cdot 0.5 = 50$.

Conditions: The coin is fair.; There are 100 tosses per trial.; $X$ counts heads.

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.

The condition must hold for *every* $\epsilon > 0$ to ensure that $f(x)$ can be made arbitrarily close to $L$. If it only held for one specific $\epsilon$, the function values might be bounded within that range but not converge to $L$.

Conditions: $\epsilon$ represents an arbitrary positive tolerance.; The definition aims to prove convergence to a specific limit $L$.

The condition must hold for *every* $\epsilon > 0$ to ensure that $f(x)$ can be made arbitrarily close to $L$. If it only held for one specific $\epsilon$, the function values might be bounded within that range but not converge to $L$.

Conditions: $\epsilon$ represents an arbitrary positive tolerance.; The definition aims to prove convergence to a specific limit $L$.

It is true because when 15 is divided by 12, the remainder is 3. This matches the definition of congruence modulo 12 used in the video.

Conditions: Use modulus 12.; Interpret the congruence via the remainder of division by 12.

It is true because when 15 is divided by 12, the remainder is 3. This matches the definition of congruence modulo 12 used in the video.

Conditions: Use modulus 12.; Interpret the congruence via the remainder of division by 12.