1. **Problem statement:** We have a binary classification problem with classes $Y = 1$ or $Y = 2$. At a data point $x$, the probability $P(Y=1 \mid X=x) = 0.4$. The nearest neighbor of $x$ is $x'$ with $P(Y=1 \mid X=x') = p > 0$. We want to find values of $p$ such that the 1-nearest neighbor (1-NN) classification error at $x$ is at least 0.5. 2. **Recall the 1-NN error formula:** The 1-NN classifier assigns the label of the nearest neighbor $x'$ to $x$. The error at $x$ is the probability that the predicted label differs from the true label at $x$. 3. **Calculate the 1-NN error at $x$: ** - The true label at $x$ is $Y=1$ with probability 0.4 and $Y=2$ with probability 0.6. - The predicted label at $x$ is the most probable class at $x'$, which depends on $p$: - If $p > 0.5$, predicted label is 1. - If $p < 0.5$, predicted label is 2. - If $p=0.5$, tie can be broken arbitrarily. 4. **Error cases:** - If predicted label is 1 (i.e., $p > 0.5$), error at $x$ is $P(Y=2 \mid X=x) = 0.6$. - If predicted label is 2 (i.e., $p < 0.5$), error at $x$ is $P(Y=1 \mid X=x) = 0.4$. - If $p=0.5$, error is at least the minimum of these two, but we consider strict inequalities. 5. **Find $p$ such that error $\geq 0.5$: ** - For $p > 0.5$, error $= 0.6 \geq 0.5$ (satisfies condition). - For $p < 0.5$, error $= 0.4 < 0.5$ (does not satisfy). - At $p=0.5$, error is ambiguous but can be considered $\geq 0.5$ if tie broken towards class 1. 6. **Final answer:** $$ p \geq 0.5 $$ The 1-nearest neighbor error at $x$ is at least 0.5 if and only if $p \geq 0.5$. This means the nearest neighbor's probability of class 1 must be at least 0.5 for the 1-NN error at $x$ to be at least 0.5.