Analyzing Decisions and Strategies with Probability - Statistics

Test indicates disease when the person is actually disease-free. Positive result despite no disease present.

$P(\text{test}+\mid\text{disease})$. Probability of positive test given disease is present.

Test indicates no disease when the person actually has the disease. Negative result despite disease being present.

$P(\text{test}-\mid\text{no disease})$. Probability of negative test given no disease.

$P(\text{disease}\mid\text{test}+)$. Probability of having disease given positive test.

Choose the option with the higher expected value. Maximizes expected gain or minimizes expected loss.

$E(X)=\sum x_i p_i$. Weighted average of outcomes by their probabilities.

$P(A^c)=1-P(A)$. Complement probability is one minus the event probability.

$P(A\cap B)=P(A)P(B\mid A)$. Joint probability equals marginal times conditional.

$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$. Updates prior probability using new evidence.

$P(\text{no disease}\mid\text{test}-)$. Probability of being healthy given negative test.

Option A, since $E_A=100\cdot^0.3=30>20=E_B$. Expected value calculation favors risky option over sure $\$20$.

Pull the goalie. Pulling doubles win probability from 6% to 12%.

$\frac{P(B\mid A)}{P(B\mid A^c)}\cdot\frac{P(A)}{P(A^c)}$. Likelihood ratio times prior odds equals posterior odds.

$P(A \cap B)=P(A)P(B)$. For independent events, joint probability equals product of individual probabilities.

$P(A \cap B)=P(A)P(B\mid A)$. Joint probability equals first event's probability times conditional probability.

$P(A\mid B)=\frac{P(A \cap B)}{P(B)}$. Conditional probability is joint probability divided by condition's probability.

$P(A^c)=1-P(A)$. Complement probability is one minus the event's probability.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Union probability adds individual probabilities, subtracts overlap.

$A,B$ independent iff $P(A\mid B)=P(A)$. Independence means conditioning doesn't change the probability.

Test negative when the person actually has the disease. False negative occurs when diseased person tests negative.

$\text{sensitivity}=P(+\mid D)$. Sensitivity is probability of positive test given disease.

$\text{specificity}=P(-\mid D^c)$. Specificity is probability of negative test given no disease.

$\text{PPV}=P(D\mid +)$. PPV is probability of disease given positive test.

$\text{NPV}=P(D^c\mid -)$. NPV is probability of no disease given negative test.

$P(D\mid +)<0.5$. Low disease prevalence makes PPV small despite high sensitivity.

$\frac{1}{3}$. Apply Bayes' theorem: $\frac{0.1(0.9)}{0.1(0.9)+0.9(0.2)}=\frac{0.09}{0.27}$.

$0.90$. Apply Bayes' theorem: $\frac{0.5(0.9)}{0.5(0.9)+0.5(0.1)}=\frac{0.45}{0.5}$.

$\frac{36}{41}$. Apply Bayes' theorem: $\frac{0.8(0.9)}{0.2(0.1)+0.8(0.9)}=\frac{0.72}{0.82}$.