Analyzing Decisions and Strategies with Probability - Statistics and Probability

1. What is sensitivity in terms of conditional probability?

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

2. What is the specificity of a test in probability notation?

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

3. What is the complement rule for an event $A$?

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

4. What is $P(D\mid +)$ if $P(D)=0.50$, $P(+\mid D)=0.90$, and $P(+\mid D^c)=0.10$?

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

5. What is the definition of independence using conditional probability?

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

6. What is the definition of a false negative in a medical test?

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

7. What is the negative predictive value (NPV) in probability notation?

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

8. What is $E(X)$ if you gain $100 with prob $0.02$ and lose $5 with prob $0.98$?

   $-\$3$. $E=0.02(100)+0.98(-5)=2-4.9=-2.9\approx-3$.

9. What is the sensitivity of a test in probability notation?

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

10. What is the conditional probability formula for $P(A\mid B)$?

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

11. What is the definition of a false positive in a medical test?

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

12. What is the positive predictive value (PPV) in probability notation?

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

13. A goalie-pull choice: If pulled, win prob is $0.12$; if not, win prob is $0.06$. Which strategy maximizes win probability?

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

14. What is $P(D^c\mid -)$ if $P(D)=0.20$, $P(-\mid D)=0.10$, and $P(-\mid D^c)=0.90$?

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

15. What is $P(D\mid +)$ if $P(D)=0.10$, $P(+\mid D)=0.90$, and $P(+\mid D^c)=0.20$?

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

16. Which option is larger if $P(D)=0.01$, $P(+\mid D)=0.99$, $P(+\mid D^c)=0.05$: $P(D\mid +)$ or $0.5$?

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

17. What is the expected value formula for a discrete outcome variable $X$?

    $E(X)=\sum x\,P(X=x)$. Expected value sums products of outcomes and their probabilities.

18. What is the definition of a false negative in a medical test?

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

19. Identify the posterior odds form: what is $\frac{P(A\mid B)}{P(A^c\mid B)}$ equal to?

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

20. Which option maximizes expected value: A pays $10 with prob $0.4$; B pays $6 with prob $0.7$?

    Option B. Option A: $E=0.4(10)=4$; Option B: $E=0.7(6)=4.2$.

21. What is the positive predictive value (PPV) in probability notation?

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

22. Which strategy has higher expected goals: pull goalie gives $0.12$ for, $0.30$ against; keep gives $0.05$ for, $0.10$ against?

    Keep the goalie in. Pull: $E=0.12-0.30=-0.18$; Keep: $E=0.05-0.10=-0.05$.

23. What is the negative predictive value (NPV) in probability notation?

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

24. Compute expected value: Option A pays $100 with prob $0.3$ else $0; Option B pays $20 for sure. Which has higher $E$?

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

25. What is the expected value formula for a discrete outcome with values $x_i$ and probabilities $p_i$?

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

26. State Bayes' rule for $P(A\mid B)$ using $P(B\mid A)$, $P(A)$, and $P(B)$.

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

27. What is the definition of a false positive in a medical test?

    Test positive when the person is actually disease-free. False positive occurs when healthy person tests positive.

28. What is the addition rule for any events $A$ and $B$?

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

29. What is the multiplication rule for independent events $A$ and $B$?

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

30. What is the general multiplication rule for events $A$ and $B$?

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

31. Which decision rule chooses the action with the larger expected value (or smaller expected cost)?

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

32. What is the multiplication rule for joint probability $P(A\cap B)$?

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

33. What is specificity in terms of conditional probability?

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

34. What is the complement rule for an event $A$?

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

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